# Quantum gravity on the black hole horizon

Nava Gaddam<EMAIL_ADDRESS>Nico Groenenboom<EMAIL_ADDRESS>Gerard ’t Hooft<EMAIL_ADDRESS>Institute for Theoretical Physics and Center
for Extreme Matter and Emergent Phenomena, Utrecht University, 3508 TD
Utrecht, The Netherlands.

###### Abstract

We study scattering on the black hole horizon in a partial wave basis, with an
impact parameter of the order of the Schwarzschild radius or less. This
resembles the strong gravity regime where quantum gravitational effects
appear. The scattering is governed by an infinite number of virtual gravitons
exchanged on the horizon. Remarkably, they can all be summed non-
perturbatively in $\hbar$ and $\gamma\sim M_{Pl}/M_{BH}$. These results
generalise those obtained from studying gravitational backreaction. Unlike in
the eikonal calculations in flat space, the relevant centre of mass energy of
the collisions is not necessarily Planckian; instead it is easily satisfied,
$s\gg\gamma^{2}M^{2}_{Pl}$, for semi-classical black holes. Therefore,
infalling observers experience no firewalls. The calculation lends further
support to the scattering matrix approach to quantum black holes, and is a
second-quantised generalisation of the same.

##  I Introduction

Inelastic collision of high energy bodies is expected to result in production
of black holes. Moreover, at large impact parameters (compared to the
Schwarzschild radius), the collapse is expected to be described by classical
physics, dominated by eikonalised single graviton exchange Banks:1999gd ;
Eardley:2002re . Of course, Hawking famously claimed that semi-classical
physics results in information loss in the final state Hawking:1976ra . This
may be seen as a thermalisation of the initial states among the large number
of degrees of freedom associated to the intermediate macroscopic black hole.
Smaller impact parameters and presumably complicated quantum corrections are
often accused of hiding the mysteries of information loss in all aspects of
scattering after collapse, in the evaporation process Giddings:2007qq .

Interestingly enough, the cross-section of the inelastic collisions producing
black holes resembles scattering of the initial states on an existing, already
formed Schwarzschild background Amati:1987wq . This is owed to the fact that,
during the collapse of a spherical shell for instance, the (apparent) horizon
forms long before the shell has fallen past the Schwarzschild radius ($R_{S}$)
of the eventual black hole. Therefore, one expects to capture a part of the
physics associated with scattering in flat space with impact parameter of the
order of the Schwarzschild radius by studying scattering on the intermediate
black hole state (already at a perturbative level). The conventional eikonal
approximation Levy:1969cr  has been used to study physics (at impact
parameters larger than $R_{S}$) in flat space gravitational scattering
tHooft:1987vrq ; Amati:1987uf ; Kabat:1992tb . In that case, the scattering
energies are necessarily Planckian $\sqrt{s}\gg M_{Pl}$ and the impact
parameter larger than the Schwarzschild radius $b\gg R_{S}$.

In the present article, we consider scattering near the Schwarzschild horizon
in a partial wave basis. Therefore, the impact parameter is necessarily of the
order of the Schwarzschild radius or less: $b\leq R_{S}$. We work in an
approximation where transverse momentum transfer on the sphere is small,
implying $b\gg L_{Pl}$. There are three scales left in the game: the centre of
mass energy of the scattering (say $\sqrt{s}$), the mass of the intermediate
black hole state $M_{BH}$, and the Planck mass $M_{Pl}$. From these, two
dimensionless parameters arise, say $\gamma\coloneqq\sqrt{8\pi
G_{N}}/R_{S}\sim M_{Pl}/M_{BH}$ and $s/(\gamma^{2}M^{2}_{Pl})$. Remarkably we
find a result that is non-perturbative in $\gamma$, by summing over an
infinite number of graviton-exchanging ladder diagrams. This implies that the
result is also non-perturbatively resummed in $\hbar$ because the ladder
diagrams include infinitely many loop contributions. At every order in
$\gamma$ in the ladder diagrams, we work to leading order in
$s\gg\gamma^{2}M^{2}_{Pl}$. However, unlike the conventional eikonal
approximation, this is far easier to satisfy for black holes much bigger than
Planck size. For a solar mass black hole, for instance, the center of mass
energy required for this approximation is $\sqrt{s}\gg 10^{-38}M_{Pl}$; this
is already satisfied for all the massive particles in the standard model with
their rest mass alone. Moreover, the energy of particles diverges near the
horizon, validating this approximation further. Therefore, this is indeed the
strong gravity regime where the mysteries of black hole information loss must
necessarily be resolved.

We compute a four-point correlation function of matter fields on the
Schwarzschild horizon. This includes multi-particle states owing to the fact
that we work with fields in a partial wave basis. The correlator captures
scattering processes in two different channels (much like in familiar nucleon-
nucleon scattering), both exchanging virtual gravitons. The result is
surprisingly simple and lends support to the S-matrix approach for quantum
black holes and generalises earlier results in Hooft:2015jea ; Hooft:2016itl ;
Betzios:2016yaq ; Betzios:2020wcv . Furthermore, this work naturally provides
a completely second-quantised description of the earlier quantum mechanical
results. While we leave the technical details to a companion article
Gaddam:2020mwe , we present an outline of the calculation, the results and
their conceptual ramifications in this one. The theory under consideration is
in Section II, the ladder diagrams to be computed in Section III, and a
discussion on the results in Section IV.

##  II The action

We consider the Einstein-Hilbert action coupled minimally to matter, say a
massless scalar field for simplicity:

$S~{}=~{}\dfrac{1}{2}\int\sqrt{-g}\,\mathrm{d}^{4}x\left(\dfrac{R}{8\pi
G_{N}}-~{}\nabla_{\mu}\phi\nabla^{\mu}\phi\right)\,.$

We focus on the fluctuations about a spherically symmetric background
$\left(g_{\mu\nu}\rightarrow g^{0}_{\mu\nu}+\kappa h_{\mu\nu}\right)$ and a
vanishing scalar profile, where

$g^{0}_{\mu\nu}~{}=~{}-2A\left(x,y\right)\mathrm{d}x\mathrm{d}y+r\left(x,y\right)\mathrm{d}\Omega^{2}_{2}\,,$

where $\mathrm{d}\Omega^{2}_{2}$ is the round metric on the two-sphere, and
$\kappa=\sqrt{8\pi G_{N}}$. For the Schwarzschild background, we have that
$A=\exp\left(1-r/R_{S}\right)R_{S}/r$ and $r$ is defined via
$xy=2R^{2}_{S}\left(1-r/R_{S}\right)\exp\left(r/R_{S}-1\right)$ in the
familiar Kruskal-Szekeres coordinates. While we will later focus on the vacuum
Schwarzschild solution in this note, we present a method that applies to
generic spherically symmetric solutions supported by matter. Of course, the
fluctuations $h_{\mu\nu}$ are not all physical and the field needs to be
gauge-fixed, an issue we now turn to.

###  II.1 Gauge fixing

The gauge that best exploits the spherical symmetry of the problem at hand is
that of Regge and Wheeler Regge:1957td . The metric fluctuations are first
expanded in spherical harmonics and split into odd and even parity modes
$h_{\mu\nu}=\sum_{l,m}h^{+}_{lm,\mu\nu}+\sum_{l,m}h^{-}_{lm,\mu\nu}$ where a
symmetric two dimensional field $H_{ab}$ with indices $a,b\in\\{x,y\\}$ and a
scalar field $K$ are introduced. Both $H_{ab}$ and $K$ are only functions of
$x,y$ (and of course depend on the suppressed $l,m$ indices) since the angular
dependence is extracted out in the spherical harmonic expansion; we label the
angular coordinates by capitalised indices $A,B$. In the Regge-Wheeler gauge,
the even mode is then given by

$\displaystyle
h^{+}_{lm,\mu\nu}~{}=~{}{\left(\begin{array}[]{cc}{H_{ab}Y_{lm}}&{0}\\\
{0}&{r^{2}g_{AB}KY_{lm}}\end{array}\right)}\,,$ (3)

and the explicit form of the odd mode will turn out to be irrelevant (as we
will describe below). Of the six physical degrees of freedom in the metric
fluctuations, four sit in the even mode above, and two in the odd-parity mode
$h^{-}_{lm,\mu\nu}$. Conveniently, the choice of gauge results in a decoupling
of the odd and even parity modes in the quadratic action. Moreover, it can be
checked Gaddam:2020mwe  that the spherical symmetry of the background ensures
a decoupling between the different partial waves at quadratic order.

Finally, a comment on Faddeev-Popov ghosts is in order. We will assume that
they contribute to the scattering to be considered only at sub-leading order
in the $s\gg\gamma^{2}M^{2}_{Pl}$ limit; this is familiar from Kabat:1992tb .
A further comment on this matter is due in Section II.2.1.

###  II.2 An effective two-dimensional theory

The spherical symmetry of the background and the decoupling between the
different parity modes allows for the theory to be reduced on the sphere to
arrive at an effective two-dimensional one Martel:2005ir . Since the two-
dimensional metric is conformally flat, a Weyl rescaling of the form
$g_{ab}\rightarrow A\left(x,y\right)\eta_{ab}$ allows for a trade of the
curvature for additional potential terms. Furthermore, the field re-
definitions $\mathfrak{h}_{ab}=rA\left(x,y\right)H_{ab}$, $\mathcal{K}=rK$
allow us to finally write the resulting effective two-dimensional action where
the space-time indices are raised and lowered by the flat metric $\eta_{ab}$
owing to the Weyl rescaling:

$\displaystyle S_{2d}~{}$
$\displaystyle=~{}\sum_{l,m}\dfrac{1}{4}\int\mathrm{d}^{2}x\left(\mathfrak{h}^{ab}_{lm}\Delta^{-1}_{abcd}\mathfrak{h}^{cd}_{lm}+\mathfrak{h}^{ab}_{lm}\Delta^{-1}_{L,ab}\mathcal{K}_{lm}+\mathcal{K}_{lm}\Delta^{-1}_{R,ab}\mathfrak{h}^{ab}_{lm}+\mathcal{K}_{lm}\Delta^{-1}\mathcal{K}_{lm}+2\phi_{lm}\left(\partial^{2}-\dfrac{\partial^{2}r}{r}\right)\phi_{lm}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.-2\phi_{lm}\left(\dfrac{A\ell\left(\ell+1\right)}{r^{2}}\right)\phi_{lm}+\gamma\,\mathfrak{h}^{ab}_{lm}\partial_{a}\phi_{0}\partial_{b}\phi_{lm}\right)\,,$
(4)

The operators appearing in this action are explicitly defined in Appendix A.
The interaction vertex a priori contains two terms with the even-parity
graviton of the form $\mathcal{K}\eta^{AB}T_{AB}$ and
$\mathfrak{h}^{ab}T_{ab}$, and a similar term containing the odd-parity
graviton. However, we first note that the four point function of interest is
always built out of two three-point vertices in this theory. We consider in-
going particles in the collision to carry momentum along the coordinate $x$
while the out-going ones along $y$. That transverse momentum (along the
angular coordinates) effects are sub-leading implies that the interaction term
contains no odd-parity graviton modes Gaddam:2020mwe . Furthermore, the
assumptions that $s\gg\gamma^{2}M^{2}_{Pl}$ and $b\gg L_{Pl}$ have two
important consequences: the first is that all diagrams containing the field
$\mathcal{K}$, and those with momentum exchange are all sub-leading at every
order in $\gamma$, while the second is that the off-diagonal components of
$\mathfrak{h}^{ab}$ do not contribute. Moreover, in the scattering process
$\phi_{\ell_{1}m_{1}}\phi_{\ell_{2}m_{2}}\rightarrow\phi_{\ell_{3}m_{3}}\phi_{\ell_{4}m_{4}}$
being considered, spherical symmetry of the background implies that the
angular momentum is either transferred across the vertex (the transfer
channel) or conserved along the vertex (the conserved channel) but not
distributed across the fields with Clebsch-Gordon coefficients; so, we label
one of the fields in the interaction vertex as $\phi_{0}$ and require it to be
with some fixed angular momentum. Those diagrams that distribute angular
momentum will introduce interactions between the various partial waves and are
consequently sub-leading Verlinde:1991iu . These facts have the consequence
that the interaction vertex simplifies to be of the form displayed in (II.2).
It is worth noting that this form ignores virtual graviton exchanges with
$\mathfrak{h}^{ab}_{\ell=0}$ in the transfer channel unless the external
particles are all in the $\ell=0$ state. Including these interactions will
also introduce interactions between the various partial waves and are
therefore expected to be sub-leading.

The propagators arising from the above action are defined by

$\begin{pmatrix}\Delta^{-1}_{abcd}&\Delta^{-1}_{L,ab}\\\
\Delta^{-1}_{R,cd}&\Delta^{-1}\end{pmatrix}\begin{pmatrix}\mathcal{P}^{cdef}&\mathcal{P}^{cd}_{R}\\\
\mathcal{P}^{ef}_{L}&\mathcal{P}\end{pmatrix}=\begin{pmatrix}\delta^{ef}_{ab}&0\\\
0&1\end{pmatrix}\delta^{2}\left(x-x^{\prime}\right)\,,$

with
$\Delta^{-1}_{abcd}\Delta^{cdef}=\delta^{ef}_{ab}\delta^{(2)}(x-x^{\prime})$
and $\Delta^{-1}\Delta=\delta^{(2)}(x-x^{\prime})$. These operators are
difficult to invert in full generality. In what follows, we will discuss the
special case of the two-dimensional surface being the horizon of the
Schwarzschild black hole.

####  II.2.1 Physics near the central causal diamond

As discussed in the Introduction, scattering processes in a black hole
background naturally capture quantum gravity effects perturbatively in
$\gamma$. In this article, we will be able to obtain results non-
perturbatively in $\gamma$. Although we shall work in Kruskal-Szekeres
coordinates in this article, the premise of the said physics is best
understood in Eddington-Finkelstein coordinates. Consider Fig. 1, which is a
slight adaptation of the original figure from Hawking:1976ra .

Figure 1: In this article, we focus on the region near the central causal
diamond Betzios:2020wcv .

The asymptotic future of all in-going wave fronts from past null infinity
$\mathscr{I}^{-}$ coincides with the asymptotic past of the wave fronts
emanating from future null infinity $\mathscr{I}^{+}$; we call this region the
central causal diamond Betzios:2020wcv .

Near this region, an apparent horizon forms well before the collapsing shell
cross the Schwarzschild radius. Therefore, strong quantum gravitational
physics which is expected in scattering with impact parameter $b\leq R_{S}$ is
evidently captured by this region. In Kruskal-Szekeres coordinates, this is in
the region where the lightcone coordinates $x,y<R_{S}$. In this approximation,
we see that the $A(x,y)\sim 1+\mathcal{O}(xy)$ and $r(x,y)\sim
R_{S}+\mathcal{O}(xy)$, to linear order. As one moves ever closer to the
diamond at Planckian distances, precise methods may be possible but this is
beyond the present paper. We focus on $L_{Pl}\ll b\leq R_{S}$.

The fact that the two-dimensional theory is conformally flat suggests that the
tensorial propagator is of the form

$\mathcal{P}^{abcd}~{}=~{}\dfrac{1}{4}f_{\ell}\left(k^{2}\right)\left(\eta^{ac}\eta^{bd}+\eta^{ad}\eta^{bc}\right)\,.$
(5)

The factor of a quarter is merely chosen for convenience, and is arbitrary. A
very lengthy calculation shows that the form factor above is given by
Gaddam:2020mwe

$f_{\ell}\left(k^{2}\right)=-\dfrac{4R^{2}_{S}}{\left(\lambda+1\right)}-\dfrac{2R^{4}_{S}k^{4}}{\left(\lambda+1\right)\left(\lambda-3\right)\left(k^{2}+R^{-2}_{S}\lambda\right)}\,,$
(6)

where we defined the constant $\lambda=\ell^{2}+\ell+1$. This propagator
deserves attention. First, we observe that it displays the familiar
ultraviolet divergences in the $k\rightarrow\infty$ limit. In the soft-limit,
which is of interest because the $s\gg\gamma^{2}M^{2}_{Pl}$ limit is dominated
by diagrams with no momentum exchange, it is finite; the horizon ensures
infrared regularity. This may be interpreted as an effective two-dimensional
mass for the graviton mode $\mathfrak{h}^{ab}$ on the horizon; it is worth
noting that the four-dimensional graviton remains massless and this is merely
a curvature artefact on the horizon. Although irrelevant for the rest of this
article, the momentum dependent term is rather curious. The $\ell=0$ mode
causes a change in sign in the term. This mode is very interesting as it
corresponds to change in black hole mass Zerilli:1970se . Worse still, the
$\ell=1$ mode has a pole; indeed, equations of motion cannot be inverted for
this mode and it can be reduced to terms containing only the odd-parity
graviton Zerilli:1970se ; Nagar:2005ea  and might be considered to affect the
centre of mass motion of the black hole. Both of these modes suggest
additional gauge redundancy. Although we have fixed the Regge-Wheeler gauge,
resulting ghosts have not been accounted for, as mentioned earlier. At sub-
leading order in the $s\gg\gamma^{2}M^{2}_{Pl}$ limit, where momentum exchange
diagrams will be of importance, these additional gauge redundancies must be
addressed.

Bringing together all the notes discussed so far, we will now focus on a given
partial wave $\ell,m$ and suppress these indices. The action (II.2) near the
horizon now reduces to

$\displaystyle S_{\text{hor}}~{}$
$\displaystyle=~{}\dfrac{1}{4}\int\mathrm{d}^{2}k\,\mathfrak{h}^{ab}\,\mathcal{P}^{-1}_{abcd}\,\mathfrak{h}^{cd}~{}-~{}\dfrac{1}{2}\int\mathrm{d}^{2}p\,\phi\left(p^{2}+\dfrac{\lambda}{R^{2}}\right)\phi$
$\displaystyle\quad~{}+\gamma\int\mathrm{d}^{2}k\,\mathrm{d}^{2}p_{1}\,\mathrm{d}^{2}p_{2}\,\delta^{(2)}\left(k+p_{1}+p_{2}\right)\mathfrak{h}_{ab}\left(k\right)p^{a}_{1}p^{b}_{2}\phi_{0}\left(p_{1}\right)\phi\left(p_{2}\right)$
(7)

for each partial wave, in momentum space. The Feynman rules for the
propagators are as follows:
feyn_propagators

and the vertex is given by feyn_vertex

$\displaystyle\fmfgraph*(100,60)\fmfleft{i8,i7}\fmfright{o2}\fmf{gluon}{i7,v1}\fmf{dashes_{a}rrow}{i8,v1}\fmf{fermion}{v1,o2}\fmflabel{$\mathfrak{h}^{ab}_{\ell
m}$}{i7}\fmflabel{$p_{2}$}{i8}\fmflabel{$p_{1}~{}.$}{o2}\fmfdot{v1}\fmfv{label=$i\gamma~{}p^{1}_{a}~{}p^{2}_{b}$,label.angle=60,label.dist=20}$

Here, we included a symmetry factor of $2$ for the scalar propagators (owing
to an exchange of external scalar legs) and a factor of $8$ for the graviton
propagator to account for the $a\leftrightarrow b$ and $c\leftrightarrow d$
symmetry. The vertex drawn here contributes to the transfer channel, where the
angular momentum is transferred across the virtual graviton. A similar vertex
with two solid scalar lines contributes to the conserved channel.

##  III The non-perturbative amplitude

The tree-level transfer channel ladder diagram i feyn_tree

(80,80) i1,i2 o1,o2 $p_{2}$i1 $p_{2}$o1 $p_{1}$i2 $p_{1}$o2 dashes_arrowi1,v1
fermionv1,o1 fermioni2,v2 dashes_arrowv2,o2 gluon,label=$k=0$v1,v2

The amplitude is straight forward to calculate and is given by
$i\mathcal{M}=4is^{2}\gamma^{2}R^{2}_{S}/\left(\lambda+1\right)=4is^{2}\kappa^{2}/\left(\lambda+1\right)$,
where we defined the Mandelstam variable that gives the center of mass energy
of the collision:
$s=-\frac{1}{2}\left(p_{1}+p_{2}\right)^{2}=p^{1}_{x}p^{2}_{y}$; this is owed
to the fact that $p^{1}_{y}=0=p^{2}_{x}$ as discussed in Section II.2. Moving
on to loop diagrams, a two loop diagram in the transfer channel is of the
following form: feyn_transfertwoloop

(200,90) i1,i2 o1,o2 dashes_arrowi1,v1 fermionv5,o1 fermioni2,v2 fermionv1,v3
dashesv2,v4 fermionv4,v6 dashesv3,v5 dashes_arrowv6,o2 gluon, tension=0v1,v2
gluon, tension=0v3,v4 gluon, tension=0v5,v6

There are six such diagrams with all possible attachments for the virtual
gravitons. It is also evident that a one-loop diagram contributes to the
conserved channel and not to the transfer channel. The solid line is not
transferred across to the bottom; it stays on top instead. This is true of all
odd-loop diagrams. Therefore, the transfer channel contributes only to the
even loop diagrams and the conserved channel only to the odd ones. In a manner
similar to that of Levy:1969cr , the general order-$n$ amplitude can be
written as

$\displaystyle i\mathcal{M}_{n}$
$\displaystyle=\left(is\gamma\right)^{2n}\int\prod_{j=1}^{n}\left(\frac{\mathrm{d}^{2}k_{j}}{\left(2\pi\right)^{2}}if_{\ell}\left(k_{j}\right)\right)$
$\displaystyle\qquad\qquad\qquad\times
I\times\left(2\pi\right)^{2}\delta^{(2)}\left(\sum_{j=1}^{n}k_{j}\right)\,,$

where $I$ contains all possible matter propagators; these can be derived
following the techniques of Levy:1969cr . Noticeably, we have inserted a
factor of $is\gamma$ for each vertex following the assumption of Section II.2
that internal vertices contributing to additional virtual graviton momenta are
sub-leading at each order in perturbation theory. After some algebra
Gaddam:2020mwe , the amplitude simplifies remarkably

$\displaystyle
i\mathcal{M}_{n}~{}=~{}i\dfrac{\gamma^{2}s^{2}\left(i\chi\right)^{n-1}}{n!}f_{\ell}\left(0\right)~{}=~{}4s\dfrac{\left(i\chi\right)^{n}}{n!}\,,$
(8)

where we have defined $\chi=-\frac{1}{4}\gamma^{2}sf_{\ell}\left(0\right)$.
The nonperturbative transfer channel amplitude is now straightforward to write
down:

$i\mathcal{M}_{\text{transfer}}~{}=~{}4s\sum_{n~{}\text{odd}}^{\infty}\dfrac{\left(i\chi\right)^{n}}{n!}~{}=~{}4is\sin\left(\chi\right)\,.$
(9)

A similar calculation returns an analogous result for the conserved channel:
$i\mathcal{M}_{\text{conserved}}~{}=~{}4s\left(\cos\left(\chi\right)-1\right)$.
The complete amplitude is therefore a sum of these two channels:

$i\mathcal{M}~{}=~{}4s\left(e^{i\chi}-1\right)\,.$ (10)

This result may also be seen as the four-point amplitude $\langle\phi_{\ell
m}\left(p_{2}\right)\phi_{0}\phi_{\ell m}\left(p_{1}\right)\phi_{0}\rangle$.
Note that the only non-vanishing component of $p_{1}$ is along the $x$
direction, which is in-going into the black hole; similarly $p_{2}$ is only
non-vanishing along $y$, which is out-going from the black hole. Therefore,
the amplitude can be written as

$\langle\phi_{\ell m}\left(p_{\text{out}}\right)\phi_{0}\phi_{\ell
m}\left(p_{\text{in}}\right)\phi_{0}\rangle\sim
p_{\text{in}}p_{\text{out}}\left(e^{i\frac{\gamma^{2}R^{2}_{S}}{\lambda+1}p_{\text{in}}p_{\text{out}}}-1\right).$

Fourier transforming the out-going field, we find that the amplitude is only
non-vanishing when

$y_{\text{out}}~{}=~{}\tilde{\lambda}\,p_{\text{in}}\quad\text{with}\quad\tilde{\lambda}=\dfrac{8\pi
G_{N}}{\ell^{2}+\ell+2}\,,$ (11)

implying that $\langle\phi_{\ell
m}\left(y_{\text{out}}\right)\phi_{0}\phi_{\ell
m}\left(p_{\text{in}}\right)\phi_{0}\rangle$ is given by

$\langle\phi_{\ell m}(\tilde{\lambda}p_{\text{in}})\phi_{0}\phi_{\ell
m}\left(p_{\text{in}}\right)\phi_{0}\rangle\,.$

This result is remarkably similar to those obtained from a first quantised
formalism in Hooft:2015jea ; Hooft:2016itl ; Betzios:2016yaq , with an
important difference in the pre-factor. At first sight, this may appear to be
an inconsistency. However, it can be checked that both results are indeed
correct. Moreover, a detailed analysis shows Gaddam:2020mwe  that restricting
the graviton fluctuations to be of the form
$\mathfrak{h}_{xx}\left(x,y\right)=\mathfrak{h}_{xx}\left(y\right)\delta\left(x\right)$
on the past horizon and
$\mathfrak{h}_{yy}\left(x,y\right)=\mathfrak{h}_{yy}\left(x\right)\delta\left(y\right)$
on the future horizon results in the factor ($\ell^{2}+\ell+1$) found in the
previous literature Hooft:2015jea ; Hooft:2016itl ; Betzios:2016yaq . The
result of the present article is therefore a generalisation that includes
arbitrary off-shell graviton fluctuations.

##  IV Discussion

In this article, we studied scattering processes with an impact parameter less
than $R_{S}$ but greater than Planck scale. This is the regime where quantum
gravity effects are important. Remarkably, the amplitude can be computed non-
perturbatively in $\hbar$ and $\gamma\sim M_{Pl}/M_{BH}$, provided that the
centre of mass energy of the collision satisfies $s\gg\gamma^{2}M^{2}_{Pl}$;
this relation is easily satisfied for large black holes. The non-perturbative
amplitude captures many-particle scattering states as it is obtained for
second quantised fields in a partial wave basis. It results in a remarkable
relation between the in-going and out-going fields (11) that is a
generalisation of the analogous result in a first quantised formalism
Hooft:2015jea ; Hooft:2016itl ; Betzios:2016yaq . This lends further support
to the scattering matrix approach to quantum black holes as a resolution to
the information paradox for the external observer. Finally, given that the
centre of mass energies of the collisions are extremely suppressed
$\sqrt{s}\gg\gamma M_{Pl}$ for large black holes, the infalling observer
experiences no drama or firewalls at the horizon.

###  IV.1 Shortcomings

We now list the limitations of this work. First, we have focussed on the
Schwarzschild horizon near the central causal diamond as depicted in Fig. 1.
However, physics further away may be incorporated by studying the classical
Regge-Wheeler potential BGP2020 . We have assumed that diagrams sub-leading in
the $s\gg\gamma^{2}M^{2}_{Pl}$ approximation can consistently be dropped at
every loop order in $\gamma$; this has been proved to be consistent for
quantum field theory in flat space Levy:1969cr . While the scalar $\phi^{3}$
theory violates this consistency Eichten:1971kd , interacting vector theories
have been shown to respect it Tiktopoulos:1971hi . A formal proof of
consistency for the present case would be an interesting technical addition to
the literature. We have assumed that contributions of ghosts are sub-leading
in the $s\gg\gamma^{2}M^{2}_{Pl}$ approximation. For eikonal graviton
fluctuations around flat space, this has been shown to be true Kabat:1992tb .
In the case at hand, the non-perturbative resummation to arrive at (10)
enforces a soft limit $k=0$ for all exchanged virtual particles. This fact,
combined with the discussion below the form factor (6) lends support to this
assumption. We have also neglected higher order terms in the graviton
fluctuations. Some of these are presumably responsible for non-
renormalisability of the theory in the UV and therefore expected to be sub-
leading in the soft limit; we hope to analyse the others in the future.
Finally, we have also assumed that all contributions arising from any
interaction between the various partial waves are sub-leading; this is natural
given the spherical symmetry of the Schwarzschild solution. An important
consequence of this is that certain spherically symmetric graviton
fluctuations $\mathfrak{h}^{ab}_{\ell=0}$ drop out of the calculations in some
loop diagrams; incorporating these would shed light on the dynamical change in
mass of the black hole.

###  IV.2 Future directions

The method proposed in this article can be generalised to include various
other standard model fields. Owing to the fact that the results are governed
by the near horizon physics, they are agnostic of the asymptotic nature of the
solution. Therefore, it may be of interest to understand the CFT calculation
of the non-perturbative amplitude (10) for small black holes in AdS/CFT. It is
also interesting to repeat the analysis for charged and rotating black holes.
The amplitude (10) may provide an additional window into understanding the
quantum chaotic nature of the spectrum of black hole microstates; it would be
very interesting to adapt the boundary condition proposed in Betzios:2020wcv
to the present second quantised description. Including transverse momentum
transfer may be tractable by including a $B$-field that is familiar from
string theory tHooft:1989tkg ; Verlinde:1991iu ; Kiem:1995iy ;
deHaroOlle:1997hx ; deHaro:1998tj . Finally, further implications for the
infalling observer would be interesting to address.

## Acknowledgements

We are thankful to Panos Betzios, Anne Franzen, Steve Giddings, Olga
Papadoulaki, and Tomislav Prokopec. This work is supported by the Delta-
Institute for Theoretical Physics (D-ITP) that is funded by the Dutch Ministry
of Education, Culture and Science (OCW).

##  Appendix A Effective two-dimensional operators

The operators appearing in the action (II.2) are:

$\displaystyle\Delta^{-1}~{}$ $\displaystyle=~{}-\partial^{2}+\eta^{ab}F_{ab}$
$\displaystyle\Delta^{-1}_{R,ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}+\dfrac{1}{2}\left(U^{c}-V^{c}\right)\partial_{c}+\dfrac{1}{2}\eta^{cd}\left(\mathcal{W}^{U}_{cd}-\mathcal{W}^{V}_{cd}\right)-A\dfrac{l\left(l+1\right)}{2r^{2}}\right)+\partial_{a}\partial_{b}+U_{(a}\partial_{b)}+\mathcal{W}^{U}_{ab}-F_{ab}$
$\displaystyle\Delta^{-1}_{L,ab}~{}$
$\displaystyle=~{}-\eta_{ab}\left(\partial^{2}-\dfrac{1}{2}\left(U^{c}-V^{c}\right)\partial_{c}-A\dfrac{l\left(l+1\right)}{2r^{2}}\right)+\partial_{a}\partial_{b}+U_{(a}\partial_{b)}-F_{ab}$
$\displaystyle\Delta^{-1}_{abcd}~{}$
$\displaystyle=~{}\dfrac{1}{2}\left(\eta_{ac}V_{[b}\partial_{d]}+\eta_{bd}V_{[a}\partial_{c]}+\eta_{ab}\left(V_{(c}\partial_{d)}+\mathcal{W}^{V}_{cd}+\dfrac{1}{2}V_{c}V_{d}\right)\right)+\eta_{ab}\eta_{cd}\left(\dfrac{1}{4}AR_{2d}-\dfrac{1}{4}V^{e}U_{e}+A\dfrac{l\left(l+1\right)}{2r^{2}}\right)$
$\displaystyle\qquad-\eta_{ac}\eta_{bd}\left(\dfrac{1}{2}AR_{2d}-\dfrac{1}{2}V^{e}U_{e}+A\dfrac{l\left(l+1\right)}{2r^{2}}\right)+\dfrac{1}{2}\eta_{cd}\left(\dfrac{1}{2}V_{a}V_{b}-V_{(a}\partial_{b)}\right)\,,$

where we defined $V_{a}=2\partial_{a}\log r$, $U_{a}=\partial_{a}\log
A\left(x,y\right)$, $\mathcal{W}^{V}_{ab}=\partial_{(a}V_{b)}$,
$\mathcal{W}^{U}_{ab}=\partial_{(a}U_{b)}$,
$R_{2d}=\frac{2}{A}\partial_{x}\partial_{y}\log A\left(x,y\right)$, and
$F_{ab}=\frac{1}{r}\partial_{a}\partial_{b}r-\frac{1}{2}U_{(a}V_{b)}+\frac{1}{4}\eta_{ab}U^{c}V_{c}$.

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Detecting Entrainment in Fermi-Bose Mixtures

Khalid Hossain
Department of Physics & Astronomy,
Washington State University, Pullman, Washington 99164–2814, USA

Subhadeep Gupta
Department of Physics,
University of Washington, Seattle, Washington 98195–1560, USA

Michael McNeil Forbes
Department of Physics & Astronomy,
Washington State University, Pullman, Washington 99164–2814, USA
Department of Physics,
University of Washington, Seattle, Washington 98195–1560, USA

We propose an experimental protocol to directly detect the Andreev-Bashkin effect (entrainment) in the bulk mixture of a bosonic and fermionic superfluid using a ring geometry.
Our protocol involves the interferometric detection of the entrainment-induced phase gradient across a superfluid due to the flow of another in which it is immersed. The choice of ring geometry eliminates variations in the stronger mean-field interaction which can thwart the detection of entrainment in other geometries.
A significant enhancement of the entrainment phase shift signal is possible, if the dimer-boson scattering length turns out to be large, which can be measured by tuning the interaction to the limit of miscibility of the two superfluids.
With suggested improvements and careful design implementation, one may achieve $\approx 67$% shift in the interferometer fringes.

§ INTRODUCTION

Superfluid entrainment plays a crucial role in the dynamics of superfluid mixtures in neutron stars.
The effect – whereby a flowing superfluid drags along or “entrains” another superfluid despite the lack of dissipation – was first predicted by Andreev and Bashkin for mixtures of superfluid ^3He and ^4He <cit.>.
Since its prediction, superfluid entrainment has been studied extensively in the contexts of neutron stars, nuclear physics <cit.>, and cold atoms <cit.>, but despite almost half a century of study, it has yet to be directly observed in experiments.

Cold atom systems provide a new environment in which to search for entrainment where one has exquisite control over both geometry and interactions <cit.>.
Entrainment manifests as an effective potential for one superfluid species induced by the flow of another. This property can be used to induce a phase winding which can then be measured using interferometry. In this paper, we outline a procedure using this effect to directly observe entrainment in a two species mixture of superfluid ^6Li and ^174Yb, taking the advantage of the large mass of the bosons and the availability of species specific potentials.

On a nuclear scale, neutron stars are cold and the crust is expected to comprise a mixture of superfluid neutrons and superconducting protons <cit.>.
These superfluids act as an internal reservoir of angular momentum, which may be suddenly transferred to the crust through vortex (un)pinning or hydrodynamic instabilities, resulting in a sudden increase in the rotation rate of the stars, observed as a “glitch” in pulsar data <cit.>.
Although the detailed mechanism behind glitches remains a mystery, it is clear that superfluid dynamics and entrainment between the proton and neutron fluids play a significant role <cit.> owing to the strength of the nuclear interactions.
Unfortunately, being light-years away, astrophysical bodies like neutron stars and pulsars are extremely difficult to measure.

For this reason we turn to ultracold atom experiments, where one can create superfluid mixtures <cit.> to act as quantum simulators of neutron-star physics <cit.>.
The challenge with terrestrial experiments is that the magnitude of the entrainment effect depends on the inter-species interaction strength $g_{ab}$, which must generally be small $g_{ab}^2 < g_{aa}g_{bb}$ to allow the superfluids to mix.
This frustrates detection in ^3He and ^4He superfluid mixtures <cit.> where the mutual interactions require temperatures below that which can be achieved by state-of-the-art cryogenics in order for the fluids to not phase separate <cit.>.
In dilute Bose gases, the entrainment is further suppressed as it occurs at second-order, depending on the square of the inter-species interactions <cit.>.
This thwarts attempts to measure the entrainment through, for example,  modifications of the dipole frequency <cit.> because mean-field effects dominate.
For the ^174Yb-^6Li- superfluid mixture considered here, the calculated value of the dipole frequency shift due to entrainment is only $\approx 0.02 \%$, two orders of magnitude smaller than the observed shift <cit.> from mean-field effects.

In addition to serving as a suitable system to observe entrainment, a mixture of fermionic and bosonic superfluids also serves as a versatile platform for other studies in many-body physics. The FFLO phase <cit.>, time-reversal-invariant superfluids with exotic topological properties <cit.>, rotational responses <cit.> and the dynamics and structure of solitons <cit.>, can be explored by tuning the mass-ratios, particle densities, and interparticle interactions. Interesting physical properties such as, the ground state characteristics of a mixture <cit.>, quasiparticle excitation spectrum <cit.>, general phase diagram <cit.>, phase competition among density wave orderings and superfluid pairings <cit.>, and many-body effects in the mixture <cit.> have been explored in great detail in the presence of optical lattice potentials, facilitating the interplay between non-linearity and periodicity. Entrainment, in particular, has been investigated theoretically for two-<cit.> and multi-component <cit.> Bose-Bose mixture in optical lattices.
By introducing an optical lattice to the superfluid mixture, one may enhance the entrainment signal by softening the phonon dispersion relationship <cit.>.

A mixture of superfluid ^6Li and ^174Yb provides a favorable environment in which to detect entrainment.
The ^6Li scattering length may be tuned with a Feshbach resonance <cit.>, allowing us to amplify entrainment effects while ensuring that the fermionic ^6Li superfluid remain mixed with the bosonic superfluid ^174Yb.
As mentioned above, although dipole oscillations provide a natural place to search for entrainment, mean-field effects dominate the frequency shift in cold atom systems.
We shall use these mean-field effects to test our microscopic theory, but then carefully design a procedure to eliminate these mean-field effects by placing the two components in ring traps.
Our proposal is similar to that suggested by <cit.>, but differs in details for removing mean-field contamination and in using interferometry to measure the effective entrainment. Another idea proposed in <cit.> is to enhance entrainment with tight confinement to induce a dimensional reduction. In contrast, with the enhancements discussed in this paper, we expect to be able to directly observe bulk entrainment in 3D.

The rest of the paper is organized as follows. In section <ref>, we derive the mean-field equations of motion for the mixture with entrainment interaction.
Section <ref> discusses the proposed experiment.
In section <ref> we estimate the phase shift.
The main results and detail discussion of the physics of the detection is presented in section <ref>: suitable experimental parameters (<ref>) and characteristics of the detectable signal (<ref>).
Finally, we conclude in section <ref>.

§ THEORY

First we consider a dilute single-component bosonic superfluid at $T=0$.
If the interactions are weak, we may describe this in the mean-field approximation by the usual GPE for the condensate wavefunction $\psi_{b}$.
This is related through a Madelung transform to the superfluid density $n(\vect{r}, t)$ and phase velocity $\vect{v} = \frac{\hbar}{m}\vect{\nabla}\phi$ where $\phi$ is the momentum potential and $\psi_{b} = \sqrt{n_{b}}e^{\I \phi}$.
The GPE follows from the following Lagrangian density:

\begin{gather}\label{eq:lagrangian1}
\mathcal{L}[\psi_{b}] = \I \hbar \psi_{b}^{\dagger}\dot\psi_{b} - \mathcal{E}(\psi_{b}),\\
\mathcal{E}(\psi_{b}) = \frac{\hbar^2\vect\nabla\psi_{b}^{\dagger}\cdot\vect\nabla\psi_{b}}{2m_{b}} + \overbrace{\frac{g_{bb}}{2} \left(\psi_{b}^{\dagger}\psi_{b}\right)^2}^{\mathcal{E}_{b}(n_{b})},
\end{gather}

where $g_{bb} = 4\pi \hbar^2 a_{bb}/m_{b}$ is the coupling constant, $a_{bb}$ is the $s$-wave scattering length between the bosons, and $m_{b}$ is the mass of the bosons.
The equations of motion give rise to the usual GPE:

\begin{equation}
\I \hbar \dot\psi_{b}(\vect{r}, t) = -\frac{\hbar^2 \nabla^2}{2 m_{b}}\psi_{b}(\vect{r}, t) + \overbrace{g_{bb}n_{b}(\vect{r}, t)}^{\mathcal{E}_{b}'(n_b)}\psi_{b}(\vect{r}, t).
\end{equation}

This formulation describes the dynamics of a weakly interacting BEC where the the equation of state for homogeneous matter is $\mathcal{E}_{b}(n_{b}) = g_{bb}n_{b}^2/2$.

However, the same formulation can be modified to describe the BEC limit of the BCS-BEC crossover of symmetric (unpolarized) fermionic superfluids with resonant $s$-wave interactions (see <cit.> for a review).
In this limit, the fermionic superfluid can be modeled with a similar equation, known as an ETF model, which describes the fermionic superfluid as a gas of condensed bosonic-dimers with number density $n_D = n_{f}/2$.
The ETF for this system is similar to the GPE with three modifications:
1) the mass is replaced by the dimer mass $m_D = 2m_{f}$ where $m_{f}$ is the mass of the fermionic components (^6Li in our case),
2) the density $n_D = \psi_D^{\dagger} \psi_D = n_{f}/2$ is interpreted as the dimer density rather than the fermion density, and
3) the interaction is replaced by the homogeneous equation-of-state in the crossover $\mathcal{E}_{f}(n_{f})$ that depends on the magnetic field $B$ through the fermion-fermion scattering length $a_{ff}(B)$:

\begin{gather}
\I \hbar \dot\psi_D(\vect{r}, t) = -\frac{\hbar^2 \nabla^2}{2 m_D}\psi_D(\vect{r}, t) + \pdiff{\mathcal{E}_{f}(2n_D)}{n_D}\psi_D(\vect{r}, t).
\end{gather}

Well justified in the BEC limit, this formulation continues to work quite well in the unitary limit where $a_{ff} \rightarrow \infty$, where it correctly describes <cit.> the hydrodynamics of rather violent collisions <cit.> and qualitative properties of vortex dynamics <cit.>.
Some of this success comes from the fact that this ETF model correctly reproduces the low-lying phonon spectrum:

\begin{gather}\label{eq:EphFermi}
E_{\mathrm{ph}}(\hbar k) = \sqrt{\frac{\hbar^2k^2}{4m_{f}}\left(\frac{\hbar^2k^2}{4m_{f}} + 4\mathcal{E}''_{f}(n_f)n_f\right)}, \qquad
k \ll k_F,
\end{gather}

where $4\mathcal{E}''_{f}(n_f) = \partial^2\mathcal{E}_{f}(2n_{D})/\partial n_D^2$, although it fails to predict pair-breaking and associated phenomena which occur near $k \sim k_F$.
This correctly reproduces the speed of sound $c_s$ where $E_{\mathrm{ph}}(\hbar k) \sim \hbar k c_s$ is linear in both the BEC and the UFG limits:

\begin{align}
c_s^2 &= \frac{g_{DD} n_D}{m_D},
&& \left.\mathcal{E}_{f} \sim \frac{-\hbar^2n_{f}}{2m_{f} a_{ff}^2} + \frac{g_{DD}n_D^2}{2}\right|_{a_{ff}\ll 1}, \tag{BEC}\\
c_s^2 &= \frac{\xi}{3}v_{f}^2,
&& \left.\mathcal{E}_{f} \sim \xi \frac{\hbar^2 (3\pi^2 n_{f})^{5/3}}{10 \pi^2 m_{f}}\right|_{a_{ff}\rightarrow \infty}, \tag{UFG}
\end{align}

where $g_{DD} \approx 0.6 a_{ff}$ is the dimer-dimer scattering length <cit.>, and $\xi = \num{0.3742(5)}$ <cit.> is the universal Bertsch parameter <cit.>, combining experiment <cit.> and QMC <cit.> values.
In our analysis, we use a Padé approximant for $\mathcal{E}_{f}(n_{f})$ that correctly interpolates between the unitary and BEC limits including QMC values for the Tan contact <cit.> and few-body parameters in the deep BEC limit <cit.>.

The Lagrangian (<ref>) is Galilean invariant under a boost of velocity $\vect{v}$ with an appropriate phase redefinition:

\begin{equation}
\psi(\vect{x}, t) \rightarrow e^{\I\phi} \psi(\vect{x} - \vect{v} t, t)
\qquad \hbar \phi = m\vect{v} \cdot \vect{x} - \frac{mv^2t}{2}.
\end{equation}

The generalization for two components – a boson $\psi_b$ and a dimer $\psi_D$ – has a similar form, but admits other terms allowed by Galilean covariance.
For the weakly interacting case, we add a term coupling the phase gradients of the two components:

\begin{gather}
\psi_{D,b} = \sqrt{n_{D,b}}e^{\I \phi_{D,b}} \qquad m_{D,b}\vect{v}_{D,b} = \hbar \vect{\nabla} \phi_{D,b}\\
\mathcal{L} = \I \hbar \big(\psi_D^{\dagger}\dot\psi_D + \psi_b^{\dagger}\dot\psi_b\big) - \mathcal{E}[\psi_D, \psi_b, \vect\nabla\psi_D, \vect\nabla\psi_b],\\
\begin{multlined}\nonumber
\mathcal{E}[\psi_D, \psi_b] = \frac{\hbar^2}{2m_D} \abs{\vect\nabla\psi_D}^2
+ \frac{\hbar^2}{2m_b} \abs{\vect{\nabla}\psi_b}^2 +\\
- \underbrace{\rho_{\mathrm{dr}}(n_D, n_b)\abs{\vect {v}_D - \vect {v}_b}^2}_{\mathcal{E}_{\mathrm{ent}}}
+ \underbrace{g_{bb}\frac{n_b^2}{2} + g_{Db}n_Dn_b + \mathcal{E}_f(2n_D)}_{\mathcal{E}_h(n_D, n_b)},
\end{multlined}\\
g_{bb} = \frac{4 \pi \hbar^2 a_{bb}}{m_b}, \qquad
g_{Db} = \frac{2 \pi \hbar^2 a_{Db}(m_D + m_b)}{m_Dm_b}.
\end{gather}

Here $\rho_{\mathrm{dr}}$ has the dimension of mass-density, as in $\rho = mn$.
$\mathcal{E}_{\mathrm{ent}}$ is the entrainment term:

\begin{equation*}
\abs{\vect{v}_D - \vect{v}_b}^2 = \frac{\hbar^2}{m_Dn_Dm_bn_b}\Biggl|\sqrt{\frac{m_b}{m_D}}\psi_b \vect{\nabla} \psi_D - \sqrt{\frac{m_D}{m_b}}\psi_D \vect{\nabla} \psi_b \Biggr|^2.
\end{equation*}

This term is manifestly Galilean invariant since it contains a difference between what is sometimes called the “superfluid velocities” $\vect{v}_i = \hbar \vect{\nabla}\phi_i/m_i$.
Note: for simple superfluids with only a single quadratic gradient term $\abs{\vect{\nabla}\psi}^2$, this superfluid velocity corresponds with the group velocity $\vect{j}_i/m_in_i$, but the presence of additional gradient terms, such as this entrainment term, changes the relationship between the phase gradients $\vect{\nabla}\phi$ and the group velocitiy.
In particular, as we shall see, entrainment allows a phase gradient to appear in a system even in the absence of a current, providing a way to detection of entrainment through interference.

Miscibility of the bosonic and fermionic superfluids – that they occupy the same physical space – is a requirement for detecting entrainment.
A necessary condition for miscibility is that the energy density be convex:

\begin{multline}
\left(\frac{\partial^2\mathcal{E}_h}{\partial n_D \partial n_b}\right)^2
\frac{\partial^2\mathcal{E}_h}{\partial n_D^2}
\frac{\partial^2\mathcal{E}_h}{\partial n_b^2}
\implies\\
\frac{\pi\hbar^2(1+\frac{m_{b}}{m_D})^2}{4m_{b}a_{bb}}
\frac{\mathcal{E}_f''(n_f)}{a_{Db}^2}.
\label{eq:miscibility}
\end{multline}

For our setup, the quantities on the left hand side are fixed, but the right-hand-side can be adjusted using the Li resonance through the $B$-field dependence of $\mathcal{E}_f(n_f)$ to ensure that our mixture remains miscible as we attempt to maximize the entrainment.

In particular, the miscibility depends sensitively on the dimer-boson scattering length $a_{Db}$.
Naïvely one might expect this to be twice the Fermi-Bose scattering-length $a_{Db} \sim 2a_{fb}$, but mean-field effects enhance this $a_{Db} \sim 3.87a_{fb}$ for our $m_{\ce{Yb}}/m_{\ce{Li}} \approx 29$ mass ratio <cit.>.
However, as pointed out in <cit.>, $a_{Db}$ may receive large in-medium corrections from three-body effects, which must be calibrated to the system under consideration.
We thus consider $a_{Db}/a_{fb}$ as an unknown parameter which must be carefully measured, and present our main results for a range of plausible values.

Miscibility condition <ref> as a function of $a_{Db}/a_{fb}$ <cit.> and magnetic field $B$.
The shaded region corresponds to immiscible fluids which cannot be used to measure entrainment.
To maximize the entrainment signal, one should generally choose the smallest magnetic field allowed by the miscibility condition (solid lines).
Within the hatched region, however, particle losses are particularly high <cit.>.
Thus if $4 \lessapprox a_{Db}/a_{fb} \lessapprox 6.5$, one should keep $B\approx \SI{730}{G}$ to maintain a reasonable lifetime of the system.

The resulting energy-density functional can be cast in terms of a wavefunction-dependent effective mass matrix for the two-component wavefunction $\vect{\Psi} = (\psi_D, \psi_b)^T$:

\begin{gather}
\mathcal{E}[\Psi] = \frac{\hbar^2}{2}\vect{\nabla}\vect{\Psi}^\dagger \cdot\mat{M}^{-1}\cdot \vect{\nabla}\vect{\Psi}
+ \mathcal{E}_h,\\
\mat{M}^{-1}[\psi_D, \psi_b] = \begin{pmatrix}
m_D^{-1} - f_{\mathrm{ent}}\frac{m_b}{m_D}n_b
& f_{\mathrm{ent}}\psi_b^\dagger\psi_D \\
f_{\mathrm{ent}}\psi_D^\dagger\psi_b & m_b^{-1} - f_{\mathrm{ent}}\frac{m_D}{m_b}n_D
\end{pmatrix},\label{eq:E}
\end{gather}

\begin{gather}
f_{\mathrm{ent}}(n_D, n_b) = \frac{\rho_{\mathrm{dr}}}{m_Dm_bn_Dn_b}.
\end{gather}

Varying the associated action with respect to $\vect{\Psi}^\dagger$ gives the following coupled non-linear Schrödinger equations:

\begin{gather}
\I \hbar e^{\I\eta}\dot{\vect{\Psi}} = -\frac{\hbar^2}{2} \vect{\nabla} \Big(\mat{M}^{-1}[\psi_{D,b}]\cdot \vect{\nabla} \vect{\Psi}\Big)
+ \mat{V}_{\mathrm{eff}}[\psi_{D,b}]\cdot\vect{\Psi},
\label{eq:DGPE}
\end{gather}

where we include a complex phase $\eta$ to simulate thermal dissipation (sometimes referred to as the DGPE or PDGPE) and the effective potential is

\begin{gather*}
\begin{multlined}
\vect{V}_{\mathrm{eff}} = \begin{pmatrix}
\pdiff{\mathcal{E}_{h}}{n_D} & 0\\
0 & \pdiff{\mathcal{E}_{h}}{n_b}
\end{pmatrix}
\alpha
\frac{\hbar^2}{2}
\begin{pmatrix}
\pdiff{f_{\mathrm{ent}}}{n_D} & 0 \\
0 & \pdiff{f_{\mathrm{ent}}}{n_b}
\end{pmatrix} +
\\
\frac{\hbar^2}{2}
\begin{pmatrix}
\frac{m_D}{m_b}\abs{\vect \nabla \psi_b}^2
& -\vect \nabla \psi_b^\dagger\cdot\vect{\nabla} \psi_D \\
- \vect \nabla \psi_D^\dagger\cdot\vect{\nabla} \psi_b
\frac{m_b}{m_D}\abs{\vect \nabla \psi_D}^2
\end{pmatrix},
\end{multlined}\\
\begin{multlined}
\alpha =
- \frac{m_b}{m_D}n_b\abs{\vect{\nabla} \psi_D}^2
- \frac{m_D}{m_b}n_D\abs{\vect{\nabla}\psi_b}^2 + \\
+ \psi_D^\dagger\psi_b\vect{\nabla} \psi_b^\dagger\cdot\vect{\nabla}\psi_D
+ \psi_b^\dagger\psi_D\vect{\nabla}\psi_D^\dagger\cdot\vect{\nabla} \psi_b.
\end{multlined}
\end{gather*}

\begin{align*}
\vect{V}_{\mathrm{eff}} =& \begin{pmatrix}
\pdiff{\mathcal{E}_{h}}{n_D} & 0\\
0 & \pdiff{\mathcal{E}_{h}}{n_b}
\end{pmatrix}
\alpha \pdiff{f_{\mathrm{ent}}}{n_D}
- \frac{m_D}{m_b}f_{\mathrm{ent}}\vect \nabla \psi_b^\dagger\cdot\vect \nabla \psi_b
& f_{\mathrm{ent}}\vect \nabla \psi_b^\dagger\cdot\vect{\nabla} \psi_D \\
f_{\mathrm{ent}}\vect \nabla \psi_D^\dagger\cdot\vect{\nabla} \psi_b
& \alpha\pdiff{f_{\mathrm{ent}}}{n_b}
- \frac{m_b}{m_D}f_{\mathrm{ent}}\vect{\nabla} \psi_D^\dagger\cdot\vect \nabla \psi_D
\end{pmatrix},\\
\alpha =&
- \frac{m_b}{m_D}n_b\vect{\nabla} \psi_D^\dagger\cdot\vect{\nabla} \psi_D
- \frac{m_D}{m_b}n_D\vect{\nabla}\psi_b^\dagger\cdot\vect{\nabla} \psi_b + \\
&+\psi_D^\dagger\psi_b\vect{\nabla} \psi_b^\dagger\cdot\vect{\nabla}\psi_D
+ \psi_b^\dagger\psi_D\vect{\nabla}\psi_D^\dagger\cdot\vect{\nabla} \psi_b.
\end{align*}

\begin{gather}
\mathcal{E}_h(n_D, n_b) =
g_{DD}\frac{n^2_D}{2} + g_{bb}\frac{n^2_b}{2}+g_{Db}n_Dn_b,\\
g_{ii} = \frac{4 \pi \hbar^2 a_{ii}}{m_i}, \qquad
g_{Db} = \frac{2 \pi \hbar^2 a_{Db}(m_D + m_b)}{m_Dxm_b},
\end{gather}

For weakly interacting bosons, $\mathcal{E}_f(2n_D) \equiv g_{DD}n_D^2/2$,
Shevchenko et. al <cit.> have calculated $\rho_{\mathrm{dr}}$ to leading order in the inter-species coupling $g_{Db}$ in terms of the phonon dispersion relationships $E_{\mathrm{ph}}(\hbar k)$.
We apply their results to our Bose-Fermi mixture by using (<ref>) for the dimers instead of the usual bosonic phonon dispersion.
The resulting expressions are rather complicated to display analytically, so we include a simple numerical implementation in the supplement <cit.>.

This approach captures the correct physics in the deep BEC limit of the crossover where $E_{\mathrm{ph}}(\hbar k)$ approaches the usual bosonic dispersion for the dimers, and provides a qualitative extrapolation to the unitary regime.
In the unitary regime, the corresponding expression neglects the softening of the phonon dispersion relationship and pair-breaking effects.
We expect this to provide an upper bound on the magnitude of the entrainment since softening the dispersion increases the number of virtual excitations that contribute to entrainment.
This modification of the dispersion relation and hence the enhancement can be achieved by using an optical lattice potential also.
The details of this mechanism are discussed in <cit.>.

Our approach starts to breakdown in the limit of strong interactions as approximating the interactions using $s$-wave scattering lengths only is not appropriate. Consequently beyond mean-field corrections arise, which should be addressed properly in the calculation.

§ PROPOSED EXPERIMENT

Using this formalism, we model a mixture of a fermionic (^6Li, ^2S_1/2) superfluid and a bosonic (^174Yb, ^1S_0) superfluid.
Using the wide s-wave Feshbach resonance centered at 832G, one can tune the fermionic $a_{ff}$ scattering length between the two lowest hyperfine states of Li <cit.>.
Since Yb is not magnetically susceptible, the $a_{bb}$ scattering length is fixed.

We propose using an optical ring trap to hold both superfluids with the ring oriented horizontally.
Several convenient values for wavelength exist above 700nm, although the polarizibility of the two species will be different (for instance at a wavelength of 780nm the polarizability ratio of Li to Yb is about 5), the minima of the potentials coincide, trapping the clouds in the same region of space, but with different trapping frequencies.
In the vertical direction, the weaker gravitational force on the Li can be countered with a magnetic field gradient to ensure physical overlap of the superfluids <cit.>.

A circular flow around the ring will then be generated in the fermionic superfluid using the procedure of optical stirring <cit.> with a laser beam which acts as a species selective potential for Lithium. A convenient wavelength is 665nm where the polarizability ratio is about 60.
Such techniques can generate circular flow with 4 windings, corresponding to a flow velocity $v/v_c \approx 0.1$ <cit.> where $v_c$ is the local speed of sound in the center of the cloud.

As discussed below, the entrainment signal is maximized at lower magnetic fields (BEC limit) where the fermionic superfluid density increases.
This can however cause two undesirable effects: the immiscibility of the fluids, and increased 3-body loss.
To mitigate these issues, we propose generating the flow at $B=\SI{832}{G}$ UFG resonance, then slowly reducing $B$ to maximize the signal once the flow is established.
The loss of miscibility at low $B$ has the interesting side-effect of allowing one to estimate the value of $a_{Db}$ by tuning the magnetic field to the limit of miscibility.

Magnetic field dependence of the phase shift for the mean-field value of $a_{Db} = 3.87a_{fb}$.
The unitary Fermi gas is realized at the resonance $B=\SI{832}{G}$ where $k_Fa_{\ce{Li}} = \infty$.
Here, we show an example calculation performed for both current and future parameters as listed in <ref>.
The hatched region is the range withing which the particle losses become significantly higher.

To prevent flow in the bosonic superfluid, a species-selective repulsive barrier seen only by the bosons can be inserted.
A convenient optical frequency for this potential is about 200GHz blue detuned of the $^{1}S_0$ $\rightarrow$ $^{3}P_1$ transition at 556nm for Yb.
After letting the two superfluids equilibrate, the bosonic superfluid will acquire a phase difference across the barrier due to the entrainment terms.
This can subsequently be measured by observing the interference pattern after expansion <cit.>.

This might seem unusual since the barrier arrests any flow in the bosonic superfluid $\vect{v}_b = 0$, so how can a phase shift accumulate?
The resolution is that entrainment modifies the relationship between the phase gradient $\vect{\nabla}{\phi}_{b/D}$ and the group velocity $\vect{v}_{b/D}$:[In this case, where the momentum operators appear in the functional (<ref>) at only quadratic order, these group velocities can be obtained simply as the derivative of the appropriately defined dispersion: $v_{b,D} = \partial E(\hbar k_b, \hbar k_D)/\partial (\hbar k_{b,D})$.
The presence of higher derivatives complicates this, and one must compute the proper momentum currents in terms of the symmetric ordered expansion of these derivatives.]

\begin{align}
\vect{j}_b &=
- \rho_{\mathrm{dr}}\left(
\overbrace{\frac{\hbar\vect{\nabla}\phi_b}{m_b}}^{\vect{v}_b}
\overbrace{\frac{\hbar\vect{\nabla} \phi_D}{m_D}}^{\vect{v}_D}
\right),\\
\vect{j}_D &=
n_D\hbar \vect{\nabla}\phi_D
- \rho_{\mathrm{dr}}\left(
\frac{\hbar\vect{\nabla}\phi_D}{m_D}
- \frac{\hbar \vect{\nabla} \phi_b}{m_b}
\right).
\end{align}

Here, we consider Yb-wavefunction (Li-dimer wavefunction), $\psi_b =\sqrt{n_b} e^{i\phi_b}$ $(\psi_D = \sqrt{n_D}e^{i\phi_D})$, and $n_b$ $(n_D)$ is the density of the bosonic (dimer) homogeneous state.

The barrier prevents any flow of the bosons, $\vect{j}_{b} = 0$, but the entrainment still allows the accumulation of a phase shift in the presence of fermionic flow $\vect{j}_{D} \neq 0$:

\begin{align}
\frac{\hbar\vect{\nabla}\phi_b}{m_b} &= - \frac{\hbar\vect{\nabla}\phi_D}{m_D}\frac{\rho_{\mathrm{dr}}}{m_bn_b}\left(1 - \frac{\rho_{\mathrm{dr}}}{m_b n_b}\right)^{-1},\\
\vect{j}_{D} &= n_D \hbar\vect{\nabla}\phi_D
\underbrace{
\left(1 - \frac{\rho_{\mathrm{dr}}}{m_Dn_D}\left(1 - \frac{\rho_{\mathrm{dr}}}{m_bn_b}\right)^{-1}\right)}_{m_D/m^*_{D}}.
\end{align}

Thus, a circulating fermionic superfluid with $\vect{j}_{D} \neq 0$ will induce a phase gradient $\vect{\nabla}\phi_b \approx -\rho_{\mathrm{dr}}\vect{j}_{D}/(\hbar n_b n_D m_D)$ in the bosons, despite the fact that the barrier keeps the bosonic cloud stationary, $\vect{j}_b = 0$:

\begin{gather}
\hbar n_b\vect{\nabla}\phi_b
= -\vect{j}_{D}\frac{\rho_{\mathrm{dr}}}{m_D n_D}
\left(1 - \frac{\rho_{\mathrm{dr}}}{m_Dn_D}
\left(1 + \frac{m_Dn_D}{m_bn_b}\right)
\right)^{-1}.
\end{gather}

Once the phase-shift has been imprinted on the bosons, and the fermions have been removed, the bosonic cloud can be released, allowing the opposite sides of the barrier to expand into each other, forming an interference pattern shifted by the relative phase imprint from entrainment.
Crucial to the success of this protocol is minimizing contamination from mean-field effects, which, as we mentioned in the introduction, can be several orders of magnitude larger than the corresponding entrainment effects, precluding observation through other methods such as the dipole frequency shift in a harmonic trap.
To mitigate this, our setup develops the phase shift in a homogeneous background around the ring geometry.
We have verified through our simulations that the back-reaction from the barrier in the bosonic cloud does not induce any mean-field effects, despite the fermionic flow.
It will be also crucial to remove the fermionic cloud with a sudden vertical laser pulse before performing the interferometery.
Repeating the experiment with circulation in the opposite direction can be used to test for and mitigate any asymmetry in the underlying trap geometry.
Numerical values for the proposed experiment are provided in section <ref>.

Entrainment phase shift ($\Delta \theta/2\pi$) calculated with magnetic field ($B$) values from the miscibility criteria (equation <ref>) for a range of values for $a_{Db}/a_{fb}$.
The top (bottom) set of curves is for the “future” (“current”) values of parameters in table <ref>.
The dashed region is the high loss region.
For those $a_{Db}$ values, we recommend $B$ = 730G, for which the mixture remain miscible.
The phase shift data is plotted in a log scale.
The points represent the phase shift values calculated using equation (<ref>) at the fiducial points in table <ref>.

§ ESTIMATED PHASE SHIFT

To estimate the phase shift, we consider a homogeneous gas with wavefunctions:

\begin{align}
\psi_D(\vect{x}) &= \sqrt{n_D}e^{\I \vect{k}_D\cdot \vect{x}}, &
\psi_b(\vect{x}) &= \sqrt{n_b}e^{\I \vect{k}_b \cdot \vect{x}}.
\end{align}

with $k_{D,b} = \nabla \phi_{D, b}$.

In our ring geometry, $k_{D} = N_w/R$ where $N_w$ is the number of phase windings, and $R$ is the radius of the ring.
Physically, the magnitude of the winding will be limited by the speed of sound in the superfluid, which defines the local Landau critical velocity $v_c$ which follows from the linearized form of (<ref>): $v_c \approx E_{\mathrm{ph}}(\hbar k)/\hbar k \approx E_{\mathrm{ph}}'(\hbar k)$:

\begin{gather}
v_D = \frac{\hbar k_D}{m_f} < v_c
\approx \sqrt{\frac{n_D}{m_D}4\mathcal{E}_f''(n_f)}.
\end{gather}

Minimizing (<ref>) we obtain the induced phase gradient in the bosonic component and the corresponding fractional phase shift

\begin{gather}
\left\lvert\frac{\Delta\theta}{2\pi}\right\rvert = \frac{2\pi R \vect{\nabla}\phi_b}{2\pi}
\vect{\nabla}\phi_D\frac{R\rho_{\mathrm{dr}}}{m_Dn_b}\left(1 - \frac{\rho_{\mathrm{dr}}}{m_b n_b}\right)^{-1}.
\end{gather}

can be as large as $67\%$, as we show in the next section.

§ SIMULATIONS

We first verify our model, by calculating the dipole oscillation frequency of the bosonic cloud in a Yb-Li superfluid mixture using the experimental parameters of <cit.>.
In this experiment, a small Yb BEC oscillates in a large ^6Li cloud at $B=\SI{780}{G}$ in a stable mixture of two-superfluids featuring large mass mismatch and distinct electronic properties.
In principle, the entrainment terms should modify the oscillation frequency from that of the background harmonic trap, but as discussed in the introduction, this shift is two orders of magnitude smaller than the shift due to the mean-field interaction.

As validation of our mean-field implementation, we reproduce the observations in <cit.>, obtaining a frequency shift of dipole oscillation consistent with the measured values. We include the trap-offset in the direction of the gravity and use $a_{Db} = 2 a_{fb}$.

In the presence of the Li cloud, the dipole oscillation frequency ($\omega_{d}$) of the Yb-BEC is extracted to be $2\pi\,\times\,\SI{381.3(4)}{Hz}$.
From our calculation, the frequency is $2\pi\,\times\,\SI{381.4(9)}{Hz}$ – well within the experimental errors.
In the experiment, along with the CM mode, scissor modes get excited.
The absence of the growth of the scissor mode shows very small energy transfer from the CM mode.
Our simulations capture the same qualitative effect.

A subtle point in the experiment was the observation of a decay in the amplitude of the dipole mode.
They conjectured that this may be due to the excitation of quadrupole modes to which the dipole mode is coupled by anharmonicity in the trapping potential.
Within our model, we find that the frequencies of the two modes are sufficiently distinct that energy cannot be efficiently transferred, and suggest instead that this dissipation is due to thermal effects.
We can reproduce the measured decay constants ($\omega_{d}\tau$) within our DGPE model (<ref>) with a phase of $\eta \approx 0.0015$. The measured and calculated values are $250$ and $210$ respectively. They have a reasonable agreement.

Aside from this effect, we reproduce the results with our self-consistent model, verifying the excitation and behavior of scissor modes from misalignment in the trap, and the mean-field frequency shifts.

This validates that our model properly captures the mean-field effects which can potentially obscure entrainment signals.
We have used this model to simulate the proposed experimental procedure to detect entrainment as described in section <ref>, including all mean-field effects and density inhomogeneities due to the trapping potentials.
An important part of this validation is that the induced flow in the fermionic superfluid coupled with the bosonic density perturbation from the barrier does not produce an asymmetry in density across the barrier.
This lack of density assymetry differentiates, for example, superfluid entrainment from mutual friction <cit.>: the latter would drag and change the mean-field densities. The main difference is, entrainment interaction couples the gradients of the phases of the two wavefunctions. Therefore, once the transient fluctuations die off, we will have a persistent flow in the fermionic component, which will induce a phase gradient on the bosons leading to a velocity difference across the barrier. Once we remove the barrier in the bosonic component and let the two parts of the cloud expand into each other, the velocity difference will result in a shift in the interference pattern. We may detect this shift by comparing against the interference fringes produced in a system without the induced phase gradient.

§.§ Experimental Parameters

We present our results in terms of two sets of parameters.
A current set of parameter values that have been demonstrated through various existing experiments, and a future set of realistic parameter values optimized to detect the entrainment effect.
We find that with current parameters, a modest phase shift occurs which lies at the bounds of current detectability, however, with future improvements, a significant phase shift will be induced that should enable the first direct detection of superfluid entrainment.

Free expansion of the central density of the bosonic component of the mixture in a periodic box, using the current parameter set (left) and future parameters (right) (see table <ref>).
Notice the asymmetry around the vertical line at $L_x = 0$: a signature of the measurable entrainment shift.
These expansion simulations are done with full 3D parameters in a tube geometry which reduces the complexity of the numerical calculations significantly while having the essential aspects of the full 3D calculation.
Consequently they do not capture the full radial expansion of the ring, but mimic the density drop.
Back to back measurements will also make the signal more comparable to the expansion of a state without asymmetry.
In the calculations here, we have used $a_{Db} = 3.87a_{fb}$ and chose the two parts of the cloud to be $\approx \SI{6}{\micro m}$ separated.
The dotted lines represent the expansion without any entrainment effect, to distinguish the amount of the shift.
The lightly shaded region in the bottom-right panel shows the amount of shift $\approx \SI{5}{\micro m}$.

*Current Parameters:

Current experiment in ring traps have demonstrated the ability to trap $N_b \approx 10^5$ bosonic atoms <cit.>, and recently, $\approx 10^4$ fermionic atoms <cit.>.
In harmonic traps $N_b = N_f \approx 10^5$ has been achieved in a Fermi-Bose mixture <cit.> and $N_f \approx 10^6 - 10^7$ has been reported for single species <cit.>.
With current technology, we expect experimental improvements to trap higher numbers of fermions in the near future.

2lAdjustable Parameters
2lFixed Parameters:

$N_{f}$:     e5e6               $m_{f}=$     6

$N_{b}$:     e5e6               $m_{b}=$     174

$B$:         Miscibility limited             $\omega_{Yb}=$     $2\pi\,\times\,$400Hz

$R$:     30200m         $a_{bb}=$     5.5nm

$a_{fb}=$     1.59nm

$\alpha=$     6.2

$v/v_c$:     0.150.2            $\xi=$     0.3705

$\Delta\theta/2\pi$:     333    ($+$ in <ref>)
$a_{Db}=$     $2a_{fb}, \SI{578}{G} $

$\Delta\theta/2\pi$:     667    ($\times$ in <ref>)
$a_{Db}=$     $3.87a_{fb}, \SI{638}{G}$

Parameters ranging from current to future values and the corresponding phase-shifts.

The main limit on the maximum density is particle loss due to 3-body processes which scale as $n^3$.
We can mitigate this to some extent by adjusting the fermion density with the magnetic field, and the boson density with the trap frequencies.
For our analysis, we fix the trap frequency for the bosons at $2\pi\,\times\,$400Hz.  This gives a lifetime for the Yb-BEC of $\approx$ 2s <cit.> with a central density $\approx \SI{340}{\micro m}^{-3} \approx \SI{3.4e14}{cm}^{-3}$ and chemical potential $\mu_b \approx \SI{65}{nK} (\mu_b/\hbar\omega_{Yb}\approx 3)$ in a 30m trap.
Excessively tight traps can heat the system, reducing the expected lifetime.
We propose using the same laser to trap both chemical species, which limits the relative polarizability ($\alpha$) between the two species.
With $\alpha = 6.2$, at $\SI{638}{G}$, the trap frequency for fermions becomes $\approx 2\pi\,\times\,\SI{5}{kHz}$, setting the lifetime of Li-dimers at $\approx$107ms with chemical potential $\mu_D \approx \SI{1.43}{\micro K}$ ($\mu_D/\hbar\omega_{Li}\approx 5)$ and central density $\approx\SI{100}{\micro m}^{-3} \approx \SI{1e14}{cm}^{-3}$.
Here, the Yb-Li scattering length remains roughly constant.[Narrow Feshbach resonances ($<\SI{1}{mG}$) have recently been observed in Yb-Li mixtures <cit.>. These do not affect our discussion.]
These densities are dilute in the sense that mean-field analyses are valid.
Standard values for relevant scattering lengths in the system are: $a_{bb}$, the boson-boson scattering length, 5.5nm <cit.> and $a_{fb}$, fermion-boson scattering length, 1.59nm <cit.>

Schematic radial expansion of a ring with an induced phase gradient in the bosonic component.
As time progresses (left to right), the density decreases because of the outward radial expansion and the fringes become wider.
(This is schematic in the sense that it is in an effective 2D geometry with translation invarience out of the page, and the visual effect is enhanced by increasing the density.)

In table <ref> we compare two sets of adjustable parameters. In the table, $v/v_c$ corresponds to the flow velocity as a fraction of Landau critical velocity in Li, $\xi$ is the Bertsch parameter and rest of the parameters have their usual meaning. The left-hand side of the first column shows typical values realized in current experiments. The right-hand side shows how these could be adjusted to maximize the entrainment signal with values that should be attainable with reasonable experimental advances.

We have calculated the percentage phase shift from the unitarity up to the BEC side of the phase diagram for the Fermi-Bose mixture by numerically integrating the equation (28) of  <cit.> at T = 0 with modified equation of state and phonon dispersion relationship for the fermionic dimers. We have also determined the parameter dependencies of the percentage phase shift from scaling analysis, and it has the following form:

\begin{multline}\label{eq:scaling}
\frac{\Delta \theta}{2\pi} \approx
\Bigl(\SI{6.2}{\percent}\Bigr)
\Big(\frac{v}{0.15v_c}\Big)
\Big(\frac{a_{Db}}{3.87a_{fb}}\Big)^{\num{2.02(1)}}
\Big(\frac{B}{\SI{638}{G}}\Big)^{\num{-3.4(3)}}\\%
\Bigl(\frac{\alpha}{6.2}\Bigr)^{\num{0.58(1)}}
\Big(\frac{R}{\SI{30}{\micro m}}\Big)^{\num{0.46(1)}}
\Big(\frac{N_f}{10^5}\Big)^{\num{0.58(1)}}
\Big(\frac{N_b}{10^5}\Big)^{\num{-0.033(1)}}\\%
\Big(\frac{m_b m_f}{m_{Yb}m_{Li}}\Big)^{\num{0.54(1)}}
\Big(\frac{m_b/m_f}{m_{Yb}/m_{Li}}\Big)^{\num{0.02(1)}}.
\end{multline}

The exponents in the equation (<ref>) demonstrate the approximate dependence of the phase shift on various parameters.
The errors in the exponents demonstrate how the exponents vary due to the full functional dependence provided in the accompanying code <cit.>.
accounts for the changes in values for the currently achievable parameters (left column in <ref>) from the possible experimental parameters implementable in future (right column in <ref>).
See <ref> for a complete tabulation of these exponents and their errors.

*Future Parameters:

From <ref> we see that the most significant signal enhancements come from increasing the dimer-boson scattering length $a_{Db}$ and decreasing the magnetic field $B$ towards the BEC limit.
Miscibility limits the enhancement of these two effects, requiring one to adjusting the magnetic field $B$ according to <ref>.
To maximize the entrainment signal, one should use the lowest magnetic field allowed by miscibility (see <ref>), but particle loss rates are particularly high in the range 650720G <cit.>.
At each magnetic field, the miscibility condition gives us values of $a_{Db}$ for which the mixture remains miscible. If the unknown $a_{Db}$ value lies within the miscible $a_{Db}$ range of 650720G, then one should keep $B \approx \SI{730}{G}$ to minimize particle loss.

Other parameters that have positive effect on the enhancement are the radius of the ring ($R$), and the induced flow velocity ($v/v_c$).
In general, by increasing each of them we can increase the signal strength, but each of them is limited by potential experimental challenges.
Perfect larger ring traps are harder to obtain as they may not remain flat at the center along the azimuthal axis.
This will lead to production of higher amount of mean-field density fluctuation which
may not completely die out within the lifetime of the metastable superfluid state.
For a single bosonic species $R \approx \SI{250}{\micro m}$ <cit.> has been achieved.
This order of magnitude may be achievable for a superfluid mixture.
In our experimental protocol, creating a substantial flow in the fermionic component increases the signal strength by a considerable amount, but is strictly limited by the number of windings we can induce set by Landau criterion.
There is also the possibility to make an improvement by increasing the induced velocity to as close as possible to the critical velocity.
Typically, only $\approx 10\%$ of the critical velocity is accessed <cit.>.
A key part of our protocol is to used the same ring trap for both species which will ensure they are trapped in the same physical space.
The centrifugal force of the rotating fluid displaces the fermionic cloud slightly, but with these parameters the displacement is insignificant $\lesssim \SI{0.014}{\micro m}$.

With the current parameter set, we may produce a phase shift up to $\approx 6\%$ (considering the mean-field enhancement of $a_{Db}$) of observable phase shift. The suggested improvements (future parameters) may increase the signal to as high as $\approx 67\%$.

§.§ Entrainment Signal

We present snapshots from the 1D time evolution of the bosonic component in a tube with the current and future parameters in figure <ref>.
These are time evolution images, which demonstrate the asymmetry in the interference fringes generated from the entrainment interaction.
In an actual experimental procedure, the interfering clouds will be imaged after a time of flight expansion after turning off the confining potential. Where the density will drop and the mean-field interaction between the bosons will reduce.
In that scenario, the fringe spacing after a long expansion time $t$ is:

\begin{equation}
\lambda_s \approx \frac{2\pi \hbar t}{m d}
\end{equation}

for initial separation $d$, as seen in the earlier experiments <cit.>.
Essentially, with long time-of-flight expansion, the fringe spacing increases linearly with time and $\SI{30}{\milli s}$ is typical for earth bound laboratory experiments. This can be increased by a factor of $3$ to $10$ in microgravity setting <cit.> as the reduced gravity will substantially increase the time-of-flight.
By employing a linear optical potential (with a laser far detuned from the Yb-resonances), one may also reduce the effect of gravity and increase the time of flight <cit.>.
Competing factors here are the initial separation and the mean-field interaction. The clouds have to be initially separated enough that, they can grow large in size before the interference.
Therefore, a point of suitable compromise has to be reached to maximize the signal in a typical laboratory experiment.
The reduction in density during the time-of-flight has another positive effect on the experimental outcome. This widens the fringes, which will make them easier to resolve.
To demonstrate this effect we perform simulations in a tube geometry, which mimics the density reduction in a radially expanding cloud. Similar effect is also visibly present in the 2D expansion of the ring in figure <ref>.

§ CONCLUSION

We propose an experimental protocol to directly measure for the bulk 3D entrainment – the Andreev-Bashkin effect – in a fermionic and bosonic superfluid mixture.
By using a ring geometry with a common trap for both species, we demonstrate that atom interferometry techniques can produce measurable phase shifts with reasonable experimental parameters, and we characterize their dependence.
An important feature of our protocol is the elimination of mean-field effects which can thwart previous attempts to measure entrainment based on shifts in the dipole oscillation frequency in a harmonic trap.
The broad Feshbach resonance allows us to adjust the interaction strength of the ^6Li superfluid to ensure miscibility, and the large mass of the bosonic ^174Yb superfluid enhances the entrainment effect.
One uncertainty is the dimer-boson scattering length $a_{Db}$, which may have significant corrections from three-body physics <cit.>.
The value of this parameter still needs careful measurement, but a significant enhancement to the entrainment signal is possible if this is large, raising an intriguing potential for developing a significant entrainment effect by tuning three-body interactions.
With current technologies and assuming mean-field values for $a_{Db}$ a 6 phase shift should be reliable.
With reasonable improvements in experimental techniques, this may be as larger as 67.

§ APPENDIX

§.§ Radial Expansion

Here we demonstrate snapshots from a schematic radial expansion of a ring in 2D in figure <ref>. This simulation is done with the current parameters in table <ref>, except we modified the density, and the width of the ring to enhance the interference effect allowing more visibility for the demonstration purpose. This demonstration reiterates the importance of time-of-flight expansion in the detection procedure. Longer time-of-flight expansion make the fringes easier to resolve by increasing the fringe width.

§.§ Deviations in the exponents

Table <ref> shows the deviation in the exponents that appear in <ref>.

Parameter     Current exponent     Future exponent

$\alpha/\num{6.2}$     0.59203+-0.00002     0.56486+-0.00003

$N_b/10^5$     $\num{-0.031+-0.003}$     $\num{-0.034+-0.003}$

$N_f/10^5$     $\num{0.59202+-0.00002}$     $\num{0.56486+-0.00003}$

$(m_b m_f)/(m_{Yb}m_{Li})$     $\num{0.5609+-0.0001}$     $\num{0.5308+-0.0001}$

$R/\SI{30}{\micro m}$     $\num{0.4391+-0.0001}$    $\num{0.4691+-0.0001}$

$B/\SI{638}{G}$     $\num{-3.18+-0.05}$     $\num{-3.67+-0.02}$

$a_{Db}/3.87a_{fb}$     $\num{2.0152+-0.0002}$     $\num{2.0279+-0.0003}$

$(m_b/m_f)/(m_{Yb}/m_{Li})$     $\num{0.02+-0.01}$     $\num{0.03+-0.01}$

Extracted parameter dependence for current and future parameter values.
These are the powers $p$ appearing in corresponding terms of <ref>.
The errors express the typical range over which the exponents vary when the parameters are changed by $\pm \SI{10}{\percent}$ about the current or future values listed in <ref>.

§.§ Equation of state: ^6Li

To perform our calculations, we parametrize the equation of state of $^{6}$Li Fermi gas.
Once we fix the scattering length ($a_{\ce{Li}}$), in terms of the external magnetic field ($B$), we use the following model:

\begin{gather}
\mathcal{E}(n, x) = f(x)\mathcal{E}_{FG}(n)\\
f(x) = \frac{bx + \xi}{\frac{x}{\xi}(b+\zeta) + 1 + \frac{18\pi b x^2}{5 a_{DD}/a_{\ce{Li}}}}
\end{gather}

with $x = 1/(k_F a)$, $\mathcal{E}_{FG}(n) = \frac{3}{5}n \hbar^2 k_F^2/2m_F$.
This Padé expansion has two following limits:

\begin{gather*}
\frac{\mathcal{E}(n)}{\mathcal{E}_{FG}(n)} = \begin{cases}
\xi - \zeta x + \order(x^2) & x \approx 0 \; \text{(Unitarity)}.\\
-\frac{5}{3}x^2 + \frac{a_{DD}}{a_{\ce{Li}}}\frac{5}{18\pi x} + \order(\frac{1}{x^{2}}), & x \rightarrow \infty \; \text{(BEC)}.
\end{cases}
\end{gather*}

Here, $a_{DD}$ is the dimer-dimer scattering length, $\zeta$ is the contact, and $\xi$ is the Bertsch parameter.

\begin{align*}
\xi &= \num{0.3742(5)}, & \zeta &\approx \num{0.901}, & a_{DD} &= 0.6 a_{\ce{Li}}.
\end{align*}

The more relevant parameter in this case, the dimer-dimer scattering length, we calculate it using $a_{DD} = 0.6a_{Li}$ <cit.>.

In the BEC limit, this reproduces the expected equation of state for a BEC of dimers with density $n_D = n/2$, mass $m_D=2m_F$, and scattering length $a_{DD}$.
The value of $b = 0.25$ is chosen to match the QMC data on the BEC side of the phase diagram.

\begin{align*}
\mathcal{E}_{BEC}(n) &= -\frac{\hbar^2}{2m_F a_{\ce{Li}}^2}n + \frac{g_{DD} n_D^2}{2} + \cdots, &
g_{DD} &= \frac{4\pi \hbar^2 a_{DD}}{m_D}
\end{align*}

To relate the magnetic field strength $B$ with the scattering length, we fit the data from <cit.>, using the model from <cit.>:

\begin{gather}
a_{\ce{Li}}^{-1} = a_{bg}^{-1}\frac{B-B_0}{(B - B_0 + \Delta) (1 + \alpha(B-B_0))}\\
B_0 = \SI{832.178498}{G}, \quad
\Delta = \SI{293.396620}{G},\nonumber \\
a_{bg} = \num{-1415.05108}~a_B, \quad
\alpha = \num{0.000406405370}, \nonumber \\
\end{gather}

§.§ Entrainment Coefficient

We have used equation (28) of  <cit.> at T = 0 as $\rho_{\mathrm{dr}}$ in our calculation. The analytic form is the following:

\begin{equation}
\rho_{\mathrm{dr}} = \int \frac{dk}{2\pi^2} k^2 \gamma_{Db}^2 \sqrt{m_Dm_b}\frac{n_D n_b \left(\epsilon_{D}\epsilon_{b}\right)^{3/2}}{\Omega_{D}\Omega_{b}\left(\Omega_{D}+\Omega_{b}\right)^{3}}.
\end{equation}

\begin{gather*}
\epsilon_{i} = \frac{\hbar^2 k^2}{2m_i}, \qquad E_i = \left(\epsilon_i(\epsilon_i + 2 g_i n_i)\right), \\
\gamma_{Db} = \frac{2\pi\hbar^2a_{Db}(m_D+m_b)}{m_Dm_b}, \qquad g_{bb} = \frac{4\pi\hbar^2a_{bb}}{m_b}, \\
g_{DD} = \frac{\partial^2\mathcal{E}_{f}(2n_{D})}{\partial n_D^2}.
\end{gather*}

\begin{equation*}
\Omega_i = \left(\frac{E_b^2 + E_D^2}{2} \pm \sqrt{\frac{(E_b^2 - E_D^2)^2}{4} + 4 \gamma_{Db}^2 n_D n_b \epsilon_{D}\epsilon_{b}}\right)^{1/2}.
\end{equation*}

We have numerically verified that, in the regions of experimental interest,

\begin{equation}
\rho_{\mathrm{dr}} \approx \int \frac{dk}{2\pi^2} k^2 \gamma_{Db}^2 \sqrt{m_Dm_b}\frac{n_D n_b}{\Omega_D\Omega_b}.
\end{equation}

follows similar scaling relationships as $\rho_{\mathrm{dr}}$, and can be used to intuitively understand the scaling relationship shown in equation (<ref>).
The essence is that, in the integral, the complicated cubic dependencies essentially drop out and we can use the simplified relationship for qualitative discussions and rough estimates of the scaling exponents.
For quantitative results, we numerically integrate $\rho_{\mathrm{dr}}$ in the thermodynamic limit for homogeneous matter with all dependencies.
To estimate the finite-size effects from the confinement of the ring, we computed the integrals as momentum sums in a finite periodic box, and find only a small correction of 1.862.5% for future and current parameters respectively.
For completeness, we write the explicit parameter dependencies below:

\begin{align*}
&\gamma_{Db}(a_{Db}, m_D, m_b),
& &n_D(N_f, m_f, R, \omega_f, \xi, B) \\
&n_b(N_b, m_b, \omega_b, a_{bb}, R),
& &\omega_f(\alpha, \omega_b).
\end{align*}

§ ACKNOWLEDGEMENTS

We acknowledge the support of National Science Foundation (NSF) under grant number 1707691 and 1306647. K.H. is grateful to Jethin Pulikkottil Jacob for many insightful discussions.