Fermi–Bose mixture in mixed dimensions

One of the challenging goals in the studies of many-body physics with ultracold atoms is the creation of a topological p x + i p y superfluid for identical fermions in two dimensions (2D). The expectations of reaching the critical temperature Tc through p-wave Feshbach resonance in spin-polarized fermionic gases have soon faded away because on approaching the resonance, the system becomes unstable due to inelastic-collision processes. Here, we consider an alternative scenario in which a single-component degenerate gas of fermions in 2D is paired via phonon-mediated interactions provided by a 3D BEC background. Within the weak-coupling regime, we calculate the critical temperature Tc for the fermionic pair formation using the Bethe–Salpeter formalism, and show that it is significantly boosted by higher-order diagrammatic terms, such as phonon dressing and vertex corrections. We describe in detail an experimental scheme to implement our proposal, and show that the long-sought p-wave superfluid is at reach with state-of-the-art experiments.


I. INTRODUCTION
The quest for the experimental realization of a chiral p x +ip y superconductor in two dimensions (2D) is gathering increasing attention because this phase exhibits Majorana modes, which are relevant for constructing faulttolerant topological quantum computers [1,2].Although a chiral p-wave superfluid has been shown to occur in the A-phase of 3 He at high pressure [3] and experiments have revealed that Strontium ruthenate (Sr 2 RuO 4 ) is a p-wave superconductor [4], the manipulation of the Majorana modes in these systems remains difficult.Therefore, the prospect to create a p-wave superfluid using ultracold atoms is very appealing because these systems allow for great control of the degrees of freedom.
Several possibilities to generate chiral superfluids have been proposed in the context of ultracold atoms in optical lattices: by using orbital degrees of freedom [5,6], spin-orbit coupling [7,8] or dipolar interaction [9,10].However, these methods either bring new problems to the experimental implementation, such as heating and ultracold chemical-reactions, or require a sophisticated optical-lattice setup and further manipulations to populate the p-orbitals.
Here, we adopt a completely different, but feasible route to produce p-wave superfluids, which consists of inducing the pairing among the 2D polarized fermionic atoms through a 3D bath of bosonic excitations.The dimensional mismatch between the fermions and the excitations that mediate their interaction leads to a huge increase of the superconducting gap, and consequently of the critical temperature for the observation of the chiral superfluid.The main advantage of our proposal is that it avoids three-body losses and dynamical instabilities (phase separation), which constitute major problems in a strongly-interacting Fermi-Bose mixture.
Mixed-dimension mixtures of two-species fermions with weak interaction were investigated previously [11,12], with the coupling between polarized fermions in 2D mediated by the particle-hole excitations of a 3D Fermi-sea background.In spite of the high stability of the Fermi-Fermi mixture, the Fermi-Bose mixture, with phonon excitations, provides much higher magnitude for the p-wave coupling between fermions.Recently, a 2D-3D mixture of fermions and bosons was considered, and the Berezinskii-Kosterlitz-Thouless (BKT) critical temperature was determined accounting for effects of retardation [13].However, many-body effects were neglected.We argue here that the proximity between the Fermi and sound velocities requires the inclusion of many-body corrections, namely the vertex ladder-diagrams and the RPA dressing of the phonon propagator [14,15].
We calculate these higher-order contributions, which are usually disregarded in the BCS treatment of conventional superconductors, and show that they significantly contribute to increase the magnitude of the anomalous p-wave gap in the Fermi-Bose mixture in mixed dimensions.In this calculation, however, we do not consider the renormalization of the pole of the Green's function, nor take into account retardation effects (the influence of the frequency of the irreducible vertex).The fermions self-energy due to the scattering of the background excitations can be neglected due to the small value of the Fermi-Bose coupling g F B , and retardation effects should not provide a relevant contribution to the vertex [16] because the singularity for pair formation must come from scattering in the Fermi-surface (Cooper instability [14,17]).The simultaneous analysis of both these effects, i.e., retardation and high-order vertex correction, is a tremendous task.Since our calculations are performed in the small momentum limit, if we would consider re-tardation, it should enhance the positive region of the vertex because correlation between the fermions leads to an even higher prediction to the critical temperature for p-wave superfluid formation (T p c ) [18].Hence, the very high value of T p c that we found due to the vertex correction is actually a lower bound, given the approximations performed.
This paper is structured as follows: Sec.II presents the system Hamiltonian for bosonic and fermionic species, whereas in Sec.III the interaction between the fermions, mediated by the bosonic excitations, is characterized.In sections IV and V, we build the BCS Hamiltonian for the 2D system and solve the associated gap equation, respectively.Higher-order corrections for the gap magnitude are evaluated in Sec.VI, and the experimental feasibility, conclusions and implications of this work are discussed respectively in Sec.VII and Sec.VIII.

II. SYSTEM HAMILTONIAN
We start by defining the Hamiltonian Ĥ = ĤB + ĤF + ĤF B , where the boson-field operators φ live in 3D, whereas the polarized fermions ψ live in 2D, (assuming = 1) with the mass of the bosonic and fermionic species given by m B and m F , and their chemical potentials by µ B and µ F , respectively.The intra-and interspecies contact repulsive interactions are characterized by the coupling constants g B and g F B , respectively.We can express the boson-field operators in terms of a discrete set of bosonic modes bq , with V the volume of the 3D space, which allows us to rewrite the bosonic part of the Hamiltonian in momentum space, To characterize the Bose-Einstein condensate, we now use Bogoliubov theory to deal with the macroscopic occupation of the zero-momentum state, that is b0 = b † 0 = √ N 0 .Neglecting higher-order fluctuations, we obtain After symmetrizing the above expression, with a sum covering half of the momentum space, and performing a Bogoliubov canonical transformation bq = u q βq − v q β † −q and b−q = u q β−q − v q β † q , where we select the real parameters u q , v q in order to have diagonal-base operators ( β, β † ) for H B , we find with the energy spectrum for the free Bogoliubov-modes excitation ω q = ξ 2 q − (g B n B ) 2 , where Applying the same set of transformations for the interspecies-interaction Hamiltonian (H F B ), and considering , where S denotes the 2D surface, we get with In Eq. ( 9), the prime symbol in the sum indicates that q = 0, and we separate the components of q = (q ⊥ , q z ), to account for momentum conservation in the plane.

III. EFFECTIVE INTERACTION
As expressed in Eq. ( 2), there is no direct interaction between the polarized fermions in H F , due to the Pauli exclusion principle.We show here, however, how an indirect interaction between fermions arises from H F B .For that, we define the effective coupling constant λ eff from the four-point function Γ = Γ(p, p , k, k ; ε, ε , ν, ν ) as follows with G 0 corresponding to the free-fermion propagator and ω = ε − ε = ν − ν.Considering the weak-coupling regime, to second order in the interaction (see Fig. 1), we obtain where D 0 (q, ω) denotes the free-phonon propagator and q ⊥ = p − p = k − k.Comparing Eq. ( 11) and Eq. ( 12), we find For low-energy processes, where the scattered fermions are kept around the 2D Fermi surface, we can assume ω ∼ 0, and Eq. ( 13) can be simplified as Hence, an effective potential λ eff = V eff (q ⊥ = |p − p|) is generated between the fermions, as a function of the momentum exchange Q between the scattered particles.In 2D real space, with coordinate R, this yields an attractive Yukawa potential between the fermionic particles in the plane, with range given by the healing length ξ = 1/ √ 2m B g B n B of the BEC.

IV. BCS HAMILTONIAN
We consider the generalized BCS-type Hamiltonian in momentum space for the fermions in the plane, with a momentum-dependent mediated interaction V eff (p, k) and µ = µ F − n B g F B .According to Eq. ( 14), we consider the interaction potential with After symmetrizing the BCS Hamiltonian properly, we apply the Bogoliubov transformation and find a new basis of operators (see App.A for details) to build the diagonal form with the energy dispersion E p = A, now we can also write the gap in terms of the mean value over this new basis, to obtain V. GAP EQUATION To solve the integral equation for a momentumdependent pairing gap in Eq. (19), it is convenient to use the 2D partial-wave expansion of the effective potential [19,20], with integer, p = |p|, k = |k|, and where we associated the angles θ p = θ and θ k = ϕ.Because we are assuming low-energy processes, with the scattered momentum close to the Fermi surface, it is reasonable to consider p ∼ k = k F in the coefficients of Eq. (20).For = 1, considering the even parity of the potential, we have where with E[X] the complete elliptic integral, K[X] the complete elliptic integral of the first kind, and X = k F ξ (see the inset of Fig. 2).Since in the weak-coupling limit one  29).Inset: harmonic = 1 of the effective potential, i.e.F(X) in Eq. ( 21), as a function of X = kF ξ.
expects that the mixing of different angular momentum will be small, we are in a position to solve the gap equation by applying the pure -type ansatz p = ( ) e i θ p in Eq. (19).That gives Analytical solutions for Max and T c can be obtained in two limiting cases: 1) T → 0, where we have the maximum gap value, and 2) T → T c , where the gap goes to zero.For the first limit, we find Then, applying the orthogonality condition given by the angular integral of equation ( 23), we eliminate the sum in to obtain where we can identify the density of states in the Fermi surface ρ 2D = m F /2π and the cut-off energy scale given by the Fermi energy of the 2D system Λ ε ∼ k 2 F /2m F .Since we consider the small-momentum regime, the fermions are scattered to states around the Fermi level.As can be seen from Table I in the experimental section, k F is very close to the healing length (ξ −1 ), which characterizes the range of the interaction potential.One can show that the induced attraction Eq. ( 17) is strongest in the p-wave channel.That means that the dominant pairing instability is in the channel with orbital angular momentum = 1, and the most stable low-temperature phase, or with highest critical temperature, has p x + ip y symmetry [11,19].We can then solve Eq. ( 24) for the maximum gap with eff (k F )/8.The vertex renormalization for two particles in vacuum allows us to express the bare coupling parameter as [21], with the reduced mass and the effective two-body scattering length a eff for a 2D-3D scattering.The latter will be a function of the original 3D scattering length a F B and of the axial confinement.That gives Considering Thus, we estimate the gap in Eq. ( 25) using For a B n

VI. HIGHER ORDER CORRECTION TO THE EFFECTIVE 2D-3D INTERACTION
The previous section shows how to optimize the gap value by manipulating the condensate density, which controls the magnitude and range of the induced potential.In addition, the importance of choosing an appropriate combination of the fermion and boson atomic masses (lighter bosonic species) to maximize the gap became clear.This issue will be further explored in Sec.VII.
By choosing the Fermi wavelength and the healing length such that ξk F ∼ 1, the Bogoliubov-sound (c s ) and the Fermi velocities (v F ) will also have close values.That requires the inclusion of higher-order diagrammatic terms in our ultracold-atoms model, which are usually disregarded in BCS studies.
In the following, we calculate the four-point function to 4th order in the interaction constant g We start with the interaction between the fermions in 2D and the "phonons" of the BEC in 3D as given by Eq. ( 9) and Eq.(10).Using the finite temperature formalism with the Matsubara Green's functions, the effective interaction between the fermions in 2D is given by with the free-fermion propagator G 0 .As seen before, the second-order expansion in the coupling g F B provides where q ≡ (k 1 − k 4 , q z ) and we applied the static limit to the Bogoliubov-mode propagator D 0 .Within a higher-order expansion, we obtain the selfenergy bubble diagram (see the details of the calculation in App.B) where we identify the static polarization-bubble diagram in 2D For , the external momenta in the Fermi surface, we can easily calculate the RPA series, which yields where we defined Replacing Eq. ( 17) by the effective potential coming from the RPA correction in Eq. ( 36), we obtain an increase in the gap magnitude, as predicted by Eq. ( 25) (see also App.B and Fig. 3).Since we consider λ 0 P 0 smaller than one, we do not expect any phase instability driven by a divergence of λ RP A eff caused by the vanishing of the denominator of Eq. (36).
The critical condition given by Eq. ( 25) can be obtained alternatively through the singularity in the effective interaction, which appears when the total vertex function is calculated in the Fermi surface, considering small total momentum of the colliding particles [17,20,23].In this case, the -th harmonic in the exponent of Eq. ( 25) will be associated with the irreducible part of the vertex.Here, we determined its = 1 projection solving the Bethe-Salpeter integral equation for the ladder-series contribution.To build the series, we start with the 4th-order vertex-correction, which reads with ω q = q 2m B q 2 + 2ξ −2 and q ≡ (k 3 − p, q z ).The first term of Eq. ( 37) is related to single-particle behavior, i.e., the scattering of real phonons, whereas the second term corresponds to virtual phonon processes.Only the latter will be relevant in our calculation, which deals with the many-body effects with the 2D momentum integration performed near the Fermi surface.
To evaluate the irreducible-vertex part around the Fermi surface, perturbation theory turns out to be insufficient and we must sum the whole ladder series of diagrams, with terms proportional to the ratio c s /v F .The resulting self-consistent vertex equation is presented and solved in the App.B, after performing a partial expansion of the effective interaction λ V eff in terms of the angular components λ(|k [ 20,23], which breaks the integral equation for the total pairing vertex to a set of decoupled algebraic equations for its partial components.Finally, we obtain the vertex correction for the component = 1 where we defined Including the correction given by Eq. ( 38) into the gap equation, according to Eq. ( 25), we get This is the main result of this paper: the inclusion of higher-order diagrams, usually neglected due to their complexity, actually increases the p-wave gap by one order of magnitude and brings it to the verge of experimental possibilities.

VII. EXPERIMENTAL IMPLEMENTATION
We now discuss the experimental feasibility of our proposal.We first examine which quantum gas mixtures are suitable to implement it, then present a scheme for a mixed-dimensional trap, and finally we summarize the experimental proposals to detect a p-wave superfluid.

A. Mixture choice
The most important criterion to choose the mixture is that the critical temperature for p-wave superfluidity T p c has to be experimentally reachable [24].As guidance, we note that BECs have been evaporatively cooled to T = 0.02T BEC c = 1 nK [25] and Fermi gases with T /T F ≤ 0.05 have been reached [26].We maximize [14] under constraints imposed by the validity of our theory and experimental constraints (γ: Euler's constant 0.57).The static approximation requires that α = v F /c s 1 [15,27].In addition, since the effective potential has been obtained within a perturbative treatment, it is necessary that γ 2 ef f < (8πγ BEC ) 1/2 .Hence, the boundaries of validity of our theoretical studies request γ BEC = a B n 1/3 B 10 −3 [13] and γ eff = a eff n 1/3 B (8πγ BEC ) 1/4 ≈ 0.4.To be in the superfluid regime we finally require T p c < T KT , where T KT is the Kosterlitz-Thouless transition temperature [28,29] with For large Y , a high mass ratio m F /m B should be selected, provided that α is chosen close to α max = 3.56 m B /m F , which maximizes we chose in the following a slightly higher value, α = 1.5 α max , which barely decreases |F(X)|, but more than doubles T F .Furthermore, a low value of γ BEC is desired and we chose a value close to its minimum.Finally, a high value of γ eff has to be achieved.In order to increase γ eff , we opt for the rather high value of n B = 6 × 10 14 atoms/cm −3 and the relatively low value of a eff = 204 a 0 , where a 0 is the Bohr radius.The motivation for choosing a large density is that T F increases with n B .On the other hand, low values of a eff are more likely available in experiments than large values, and they can be reached without Feshbach or confinement induced resonances.Far from the resonances, the scattering length is given approximately by a eff ∼ m B /m F B a FB [21,30,31].
Further limitations arise from experimental constraints.In our scheme, a few thousand fermions will be sympathetically cooled by a much larger bath of evaporatively cooled bosons.To effectively implement evaporative and sympathetic cooling, a sufficient rate of elastic collisions and low rates of heating and loss are required.These conditions limit the range of suitable interaction properties, the gas densities, and the trap designs.An upper limit on n B is imposed by the requirement to keep the BEC in the 3D regime for the finite number of bosons available.A lower limit on a B is imposed by the requirement of a sufficient elastic collision rate between bosons Γ el,B ∝ n B a 2 B .Together, these requirements lead to an additional, experimental, lower limit on γ BEC .Attention has also to be given to the rate of 3-body losses involving one fermion and two bosons (Γ FBB ∝ n 2 B a 4 FB [32,33]), even considering the important role played by the mixed dimensionality in inhibiting the interspecies molecular formation [11].
We now discuss possible choices of elements for the mixture.Since m F /m B should be large, we limit our choice of bosons to the lightweight isotopes that have been Bose condensed, 4 He * , 7 Li, and 23 Na.Among those, 7 Li has the great advantage of possessing a broad Feshbach resonance, with which a B can be tuned [34][35][36][37].Feshbach resonances in 4 He * and 23 Na are expected or known to be accompanied by strong losses [38][39][40][41][42].In the following, we use the triplet-scattering length for 4 He * and 23 Na [43,44].Considering BEC densities for which inelastic collisions limit the BEC lifetime to 10 s [45][46][47], fermion masses up to the mass of the heaviest naturally occurring fermionic isotope ( 235 U) and a eff = 600 a 0 , we find that T p c /T F < 10 −2 for these bosons.Only larger values of a eff might make them suitable for our purposes.a) Figure 4. Maximum p-wave superfluid critical temperature T p c /TF (upper panels, solid lines) and T p c (lower panels, solid lines) for fermions immersed in bosonic 7 Li, as well as TKT/TF (upper panels, dotted lines) and TKT (lower panels, dotted lines).a) Dependence on the mass of the fermions mF .Here nB = d × 10 14 atoms/cm 3 , aB = 8 a0, a eff = 200 a0 (corresponding to γBEC = 0.002 d 1/3 and γ eff = 0.05 d 1/3 ), and α = 1.5 αopt.Fermionic isotopes of elements that have been cooled to quantum degeneracy are marked by vertical lines.b) Dependence on a eff for the fermion 171 Yb, with all other parameters as before.The dashed lines in the upper panels mark the experimentally achieved T /TF .The stars mark the example detailed in Table I We therefore limit our considerations to 7 Li.This choice makes it possible to decrease a B and thereby in-crease T p c /T F .To choose the fermionic element we plot in Fig. 4a) T p c /T F and T p c as a fuction of m F .Fermionic isotopes that have been cooled to quantum degeneracy and for which the experimentally relevant regime T p c /T F > 0.05 can be reached are 171,173 Yb, 161 Dy, and 167 Er [48][49][50][51][52][53].A drawback of having to choose such heavy elements could be that they are not well sympathetically cooled by the lightweight Li because during each elastic collision, the energy transfer from the fermion to the boson is suppressed by 4m F m B /(m F + m B ) 2 ∼ 0.15 [54].A benefit of Dy and Er compared to Yb is that several interspecies Feshbach resonances will likely be available across the broad 7 Li Feshbach resonance, making it possible to tune a B and a FB somewhat independently and to access large values of a FB , which would also make tuning of a eff by confinement induced resonances possible.
Nevertheless, since 173,174 Yb-6 Li mixtures are already available in the lab [55,56], we concentrate our discussion now on 171,173 Yb-7 Li.Adapting the existing machines to operate with 7 Li instead of 6 Li should be straightforward.There are two fermionic Yb isotopes readily available, each providing a chance of possessing suitable interspecies interaction properties with 7 Li. Figure 4b) shows the dependence of T p c /T F and T p c on n B and a eff .Choosing a B = 8 a 0 leads to the system parameters given in Table I.The dotted lines in Fig. 4 are an estimation of the Kosterlitz-Thouless transition temperature, which is given by [28,57] where m and n are the mass and density of the superfluid species, while a characterizes the range of the interaction.
In particular, for our case of fermionic-pair formation, the interaction between fermions that will form the Cooper pairs is proportional to a 2 F B , with m = 2m F and n ∼ n F /2. Eq. ( 41) is valid for small interaction parameters a B and a F B -the first makes the range of the potential long enough, such that the superfluid fraction achieves its maximum value [12,13].
The critical temperature T p c = 0.07 T F = 9.5 nK is in the regime of temperatures that have already been achieved experimentally, albeit in systems with larger elastic scattering length.However, T p c /T BEC c = 5 × 10 −4 is more than one order of magnitude lower than what has been reached so far.To enhance evaporative cooling, it might be useful to first evaporate at a scattering length above 100 a 0 and to tune the scattering length to a lower value only when approaching the required low temperature, while compressing the gas at the same time.In doing so, one could even profit from a Li 3-body recombination minimum at a B = 119 a 0 [35].

B. Trap configuration
Next, we consider suitable trap configurations for the mixture.Whereas the bosons explore a 3D trap, the fermions have to be effectively confined in 2D by a harmonic trap of frequency ν ⊥,F , which requires hν ⊥,F − E F k B T .The sample should be as homogeneous as possible to avoid inhomogeneous broadening of p-wave superfluidity signals, especially because the number of fermions will be low.Efficient evaporative cooling of the bosons should be possible in order to reach low temperatures.We now take these requirements into account to design an optical dipole trap for the mixture, where we orient the 2D plane of the fermions in the horizontal direction, see Fig. 5a.
The bosonic lithium surrounds the fermions and can be confined by a Gauss-beam dipole trap using a wavelength of 1064 nm.To reach a temperature T by evaporation, the trap depth in the vertical direction U ⊥,B should be µ BEC +ηk B T , where µ BEC is the chemical potential of the BEC, and η ∼ 5 [60].In order to provide a homogeneous vertical trap frequency across the cloud, the horizontal waist should be much larger than the cloud and the vertical Rayleigh length z R much longer than the horizontal sample size.The latter requirement and the additional requirement hν ⊥,B µ BEC are only fulfilled if the vertical waist is larger than a minimum size.At the same time, the vertical waist should not be too large in order to limit the size of the 7 Li sample in the vertical direction, thereby reducing the required number of 7 Li atoms.Gravitational sag of the bosonic cloud is compensated by placing the focus of the Gauss beam slightly above the plane of the fermions.The Gaussian-beam trap creates a nearly constant potential on the fermions, since they explore only a small region in the centre of the trap.A constant potential offset is irrelevant and we can therefore ignore the influence of the Gauss-beam dipole trap on the fermions.
To provide homogeneous confinement for bosons and fermions in the horizontal plane, repulsive dipole trap walls can be erected around the sample using vertically propagating Gauss beams [62,63].Four such beams can form a rectangular box with a size of ∼ 10 µm around the sample, if the waist of the beams is elongated along the sides of the rectangle (w Lh, of a few 10 µm) and is narrow orthogonal to that direction (w Lh,⊥ ∼ 2 µm).This rectangular potential box also serves to select the most homogeneous central region of the traps that are used to confine bosons and fermions vertically.The sample density can easily be changed by moving the vertical walls towards each other, which is useful to do while a B is reduced to a low value.If in further studies a cylindrically symmetric system is required, for example to enable the creation of vortices [64], a Laguerre-Gaussian beam can be used to confine the atoms horizontally [63,65,66].
The confinement of the fermions in quasi-2D is most conveniently done using optical lattices.In comparison to other trap configurations, such as a Hermite-Gaussian beam [62,67], it is easier to create a more homogeneous confinement in the 2D plane by increasing the diameter of the lattice beams.In order to populate only a single plane of the lattice with fermions, one can use the techniques of Refs.[68][69][70][71].
The deep dipole potential used to confine the fermions in 2D may only have a negligible effect on the bosons.The parasitic potential on the bosons U lattice,B must be much smaller than µ BEC .This challenge has been met by species-specific dipole traps using a "tune-out" wavelength, for which the AC polarizability of one species is zero [30,31,72,73].Unfortunately, this technique does not work for 7 Li because its "tune-out" wavelength is too close to an atomic transition, leading to detrimental off-resonant scattering for the required trap depths [72].Another option is to use a "tune-in" wavelength, close to an Yb transition and far detuned from any Li transition [72].In this situation, the potential on Yb U lattice,F can exceed the potential on Li many times.This technique is suitable for our situation, but will limit the lifetime of the fermionic cloud to a few seconds by off-resonant scattering.If this limit is significant depends on the other factors limiting the lifetime of the system, especially the unknown 3-body loss rate Γ FBB .
If the lifetime limit imposed by a "tune-in" lattice is too severe, a bichromatic dipole trap can be used, consisting of two optical lattices that both confine Yb, but compensate each other for Li.This technique overcomes the possibly excessive off-resonant scattering and replaces it by the technical challenge of creating two lattices with very well controlled intensity profiles.We will explore  this scheme in the following.We chose optical lattices with wavelengths of 470 nm and 1064 nm, which are both Table II.Optical dipole trap configuration.λLi is the wavelength of dipole-trap beam Li, with i = 1, 2. w are the 1/e beam radii.The vertical trap depth for 7 Li, U ⊥,B , takes the effect of gravity into account.αLi is the angle between lattice beams Lia and Lib.∆z is the lattice spacing.n2D,B is the density of bosons integrated over the vertical direction.τB,F = 1/ i Γi,B,F are limits to the lifetimes of bosons and fermions, where Γi,B,F is the off-resonant scattering rate of photons calculated at peak intensity of dipole trap beam Li, with i running over all beams [24,58,74].attractive for Yb.In contrast, for Li only the 1064-nm lattice is attractive, the other is repulsive, see Fig. 5b.In order for the lattice potentials to add up for Yb and cancel sufficiently for Li, the intensity profile of both lattices need to be nearly identical in the region of the atomic clouds.The lattice-well spacing must be the same, and the intensity maxima need to overlap.The lattice spacing can be adjusted by the angle between the two lattice beams of each wavelength.Using an angle of 60 • between the two beams forming the 1064-nm lattice leads to a lattice spacing of 1064 nm.The same spacing is reached for the 470-nm lattice if the two corresponding beams intersect at an angle of 25.5 • , see Fig. 5a.The position of the intensity maxima along the lattice direction (the vertical direction) depends on the phase difference between the two beams forming a lattice.This phase difference has to be stabilized interferometrically for each lattice to a common reference, combining methods from Refs.[75,76].In order for the two lattice potentials to cancel for the bosons, the intensity of the 470-nm lattice beams has to be 1.8 times the intensity of the 1064-nm lattice beams.For Yb the two lattice potentials add up, giving a total potential that is 1.2 times larger than the potential of the 470-nm lattice alone.This total potential needs to confine Yb in quasi-2D and be also deep enough to suppress tunneling of Yb to neighboring lattice planes, see Fig. 6.
The cancelation of the lattice potential for the bosons will not be perfect because of intensity and phase fluctuations leading to deviations from the ideal configuration.Phase fluctuations of 90 mrad or intensity imbalances of 9% lead to a residual potential on the order of 10% of µ BEC .This parasitic potential would be tolerable if the timescale of fluctuations is large enough to avoid heating of the sample.In principle, we could have chosen a wavelength for L2 that is further away from the Yb transition, e.g.532 nm, which would reduce off-resonant scattering and simplify phase locking of the laser sources used for L1 and L2.All the same, we chose 470-nm because at that wavelength we are profiting from less parasitic potential of L2 on Li, reducing the amount of compensation needed from L1.As a result, the overall parasitic potential created for a given intensity or phase mismatch between L1 and L2 is reduced.
Example parameters for the bichromatic dipole trap and important results of using this trap for the Li-Yb mixture are given in Table II.The 7 Li atom number available in current experiments (3 × 10 5 atoms [37]) is sufficient for a square sample of 10 µm size.A sample of this size contains about 700 fermions.If this proposal is realisable depends to a large extend on the unknown elastic and inelastic scattering properties of Li-Yb.Similar schemes can be applied to other mixtures, such as Li-Dy or Li-Er, for which some interspecies interaction tuning should be possible.

C. Detection of p-wave superfluidity
There are some predictable signatures for the experimental detection of the p x + ip y superfluid phase.Particularly, the density of state (rf absorption spectrum) of a rotating weak pairing p x + ip y phase is expected to exhibit a set of gapless modes [77], which are a direct consequence of the zero-energy Majorana modes on the vortices.The rf-spectroscopy can be also applied to detect Majorana edge states of the topological superfluid in a 2D square lattice [78].On the other hand, the timereversal symmetry broken signature of the chiral p x + ip y fermionic superfluid can be detected with time-of-flight image of the atomic density distribution: an external effective electric field (i.e., dipole interaction between the neutral atoms in the superfluid and the laser field) brings a nonzero antisymmetric transverse mass current in the velocity distribution of the atoms [79].

VIII. CONCLUSION
In the present work, we explored the feasibility of a p-wave superfluid by using a Fermi-Bose mixture in a mixed-dimension configuration, where p-wave interaction between spin-polarized degenerate fermions in 2D is induced indirectly, through the scattering of the Bogoliubov modes of condensed bosons moving in 3D.We have shown that, even in the weak-coupling regime, the appropriate renormalization of the phonon propagator (BEC modes) with particle-hole fluctuations and the vertex correction significantly increase the gap and the predicted critical temperature for the fermion-pair formation.
It is important to remark that we adopt a minimum value for γ BEC ∼ a B n 1/3 B , which yields υ F /c s ≤ 1, thus allowing to disregard retardation effects.According to Wu and Bruun [13], who performed calculations including retardation but no vertex correction to determine T M F , in the limit υ F /c s ≤ 1, it holds that T M F ∼ T BCS (see Fig. 2 in the cited reference), which confirms the validity of our approximation.
We neglected decay of the BEC phonons, like the Beliaev damping and the lifetime due to the scattered particle-hole pairs of the degenerate fermionic sample.The Beliaev damping is given by the boson-boson scattering potential, resulting in a phonon lifetime proportional to g B [80,81].In the small-momentum regime, however, the Beliaev decay mechanism is strongly suppressed [80].On the other hand, if we consider the phonon dressed by particle-hole fluctuations of the Fermi sea, it will have a lifetime proportional to g 2 F B .In the static limit considered in the paper, however, the lifetime is infinite (see App. B for details).Hence, we conclude that there is no damping mechanism that could hamper the stability of the BEC in the chosen regime of parameters.
Exploiting the difference in polarizability and mass of the atomic species, and by optimizing the density n B and the scattering length a B of the bosons, our work sets the boundary for the experimental realization of a p-wave superfluid within the reachable limit of T p c = 0.05T F .It identifies a realistic route and provides the details to the accomplishment and manipulation of this long-sought fascinating chiral-superfluid phase in the realm of ultracold atoms in optical lattices.Now we will solve the "polarization bubble" in 2D Before we integrate in momentum space, we simplify the above expression by changing the variable in the second term to p = p + k.We then obtain Since we are interested in the zero-temperature limit, we consider the analytic continuation iν → ν + iδ, with n F ( p ) → Θ(µ − ε p ).Then, we focus on the real part of Eq. (B9) Starting with the angular integral in Eq. (B10) (for |k/2k F ± νm F /kk F | > 1), after changing the variable p → ε = p 2 /2m F in the resulting integral, we obtain (see Ref. [82]) remembering that µ = k 2 F /2m F .Particularly, in the static limit ν = 0, we will have and Assuming |k 1 − k 4 | < 2k F , we can easily calculated the RPA series, which gives where we defined λ 0 = −2g 2 F B n B m B / |k 1 − k 4 | 2 + 2ξ −2 and P 0 = −m F /2π.For λ 0 P 0 < 1, we find Now, we consider the RPA correction to calculate the projected component = 1 of the potential V eff , i.e., and then with Y = V 0 ρ 2D ξ/ √ 2, and where Π[X, Y ] is the complete elliptic integral of the third kind.One can estimate the RPA gap correction comparing the minima in Fig. 3, which shows the profile of F(X) and I(X, Y ) in a broad range of X (Y ∼ 0.05/X, since we consider n B as the only tunable parameter).

Figure 1 .
Figure 1.Second-order Feynman diagram for the interaction between two fermions in 2D induced by the Bogoliubov modes of the 3D BEC.

Figure 2 .
Figure2.Profile of the function F(X)/X used to estimate the maximum gap in Eq. (29).Inset: harmonic = 1 of the effective potential, i.e.F(X) in Eq. (21), as a function of X = kF ξ.

2 Figure 5 .
Figure 5. Mixed-dimensional optical dipole trap.a) Beam configuration.Ytterbium is confined in a 2D plane of an optical lattice formed by two standing waves created by laser beam pairs L1a,b and L2a,b.Both standing waves have the same intensity profile near the trap centre and are attractive for Yb, but generate opposite potentials for Li.Lithium is confined vertically by an elliptical Gaussian beam (Lv), elongated in the out-of-plane direction.Both elements are horizontally confined by four repulsive dipole-trap walls (Lha,b,c,d), forming a rectangular box.The inset shows the region around the trap centre, with Lha,b in cross section and the lattice intensity profile.b) Dipole potential and scattering rate for Li and Yb, as a function of the wavelength [58, 59].The arrows above the graph indicate the wavelengths of the dipole-trap beams.Two choices are possible for Lh.

Figure 6 .
Figure 6.Optical dipole trap potential.A lattice confines Yb in 2D, whereas Li is levitated against gravity by a Gaussian beam.The potential experienced by thermal atoms U Li,thermal consists of the dipole potential and twice the BEC mean-field potential [61].A phase fluctuation of a lattice beam by 0.1 rad leads to the modulated Li potential shown around the ideal potential.

Table I .
[34,35]eters of171,173Yb-7Li mixture.The elastic scattering rate Γ el,B is given for thermal atoms at a temperature of T = T p c colliding with a BEC at density nB.Γ 3−body,B = − ṄB/NB is the initial 3-body loss rate of the BEC[34,35].