Spin-exchange-induced exotic superfluids in a Bose-Fermi spinor mixture

TL;DR

This paper proposes a method to realize spin-orbit coupled Fermi gases and exotic superfluid phases using a Bose-Fermi mixture, where fermions acquire effective spin-orbit coupling via interactions with spin-orbit coupled bosons.

Contribution

It introduces a novel approach to induce spin-orbit coupling in fermions through boson-fermion interactions, enabling topological superfluids without Raman heating and enhancing observable density stripe phases.

Findings

Fermions gain effective spin-orbit coupling via spin-exchange interactions.

The presence of fermions enhances the bosonic stripe phase and increases its spatial period.

The system offers a practical platform for exploring spin-orbit physics and topological phases.

Abstract

We consider a mixture of spin-1/2 bosons and fermions, where only the bosons are subjected to the spin-orbit coupling induced by Raman beams. The fermions, although not directly coupled to the Raman lasers, acquire an effective spin-orbit coupling through the spin-exchange interaction between the two species. Our calculation shows that this is a promising way of obtaining spin-orbit coupled Fermi gas without Raman-induced heating, where the long-sought topological Fermi superfluids and topological bands can be realized. Conversely, we find that the presence of fermions not only provides a new way to create the supersolid stripe phase of the bosons, but more strikingly it can also greatly increase the spatial period of the bosonic density stripes, and hence makes this phase directly observable in the experiment. This system provides a new and practical platform to explore the physics of spin-orbit coupling, which possesses a dynamic nature through the interaction between the two species.

Equations

$$\mathbf{H}=\int dx\left({\Psi }_B^{†}{h}_B{\Psi }_B+{\Psi }_F^{†}{h}_F{\Psi }_F\right)+{\mathcal{G}}_B+{\mathcal{G}}_F+{\mathcal{G}}_{BF},$$

$$h_B$$

$$h_F$$

$${\mathcal{G}}_B=\int dx\left[g^B\left({\rho }_{B\uparrow }^2+{\rho }_{B\downarrow }^2\right)+2g_{\uparrow \downarrow }^B{\rho }_{B\uparrow }{\rho }_{B\downarrow }\right],$$

$${\mathcal{G}}_F=g^F\int dx{\psi }_{F\uparrow }^{†}{\psi }_{F\downarrow }^{†}{\psi }_{F\downarrow }{\psi }_{F\uparrow },$$

$${\mathcal{G}}_{BF}=\int dx\left[\gamma {\rho }_B{\hat{\rho }}_F+\beta \text{(ψB↓∗ψB↑ψF↑†ψF↓+h.c.)}\right],$$

$$\frac{\Psi_{B}}{\sqrt{n_{B}}}=\left[C_{1}\left(\begin{array}[]{c}\cos\theta\\ -\sin\theta\end{array}\right)e^{ik_{B}x}+C_{2}\left(\begin{array}[]{c}\sin\theta\\ -\cos\theta\end{array}\right)e^{-ik_{B}x}\right],$$

$$\frac{{\mathbf{E}}_B\left(k_B,\theta ,B\right)}{N_B}=$$

$$-F(B){\cos }^2\left(2\theta \right)+G_1\left(1+2B^2\right),$$

$$\int dx{\Psi }_F^{†}{h}_F^{\mathrm{eff}}{\Psi }_F=\int dx{\Psi }_F^{†}{h}_F{\Psi }_F+{\mathcal{G}}_{BF}.$$

$$h_{F}^{\rm eff}\left(k_{B},\theta,B\right)=\frac{\left(k-k_{r}\sigma_{F}^{z}\right)^{2}}{2m_{F}}+n_{B}\left(\begin{array}[]{cc}\gamma V&-\beta M\\ -\beta M^{*}&\gamma V\end{array}\right),$$

$$M$$

$$V$$

$${\rho }_B\left(x\right)=n_B\left[1+\sin \left(2\theta \right)\cos \left(2k_{B}x\right)\right],$$

$$h_{\mathrm{F},\mathrm{PW}}^{\mathrm{eff}}=\frac{{\left(k-k_r{\sigma }_F^z\right)}^2}{2m_F}+\frac{{\Omega }_F^{\mathrm{eff}}}{2}{\sigma }_F^x,$$

$${\Omega }_F^{\mathrm{eff}}=-\beta n_B\sin \left(2\theta \right).$$

$$e^{i{Z}_j}=\stackrel{[}{a}=-d]d\prod w_j^{\ast }\left(k_a\right)\cdot w_j\left(k_{a+1}\right),$$

$${\mathbf{P}}_F\left(\mu ,\theta \right)=\int dx{\Psi }_F^{†}\left(h_{\mathrm{F},\mathrm{PW}}^{\mathrm{eff}}-\mu \right){\Psi }_F+{\mathcal{G}}_F,$$

$${\mathcal{G}}_F=-\Delta \left[{\psi }_{F\uparrow }^{†}\left(x\right){\psi }_{F\downarrow }^{†}\left(x\right)+h.c.\right]-{\Delta }^2/g^F,$$

$$\Delta =-g^F⟨{\psi }_{F\downarrow }\left(x\right){\psi }_{F\uparrow }\left(x\right)⟩.$$

$${\mathbf{P}}_F\left(\mu ,\theta ,\Delta \right)=\frac{1}{2}\sum _k{C}_k^{†}{M}_k{C}_k+\sum _k{\xi }_k-\frac{L{\Delta }^2}{g^F},$$

$$M_{k}=\left[\begin{array}[]{cccc}\xi_{k}+\lambda k&\Omega_{{\rm F}}^{{\rm eff}}/2&0&-\Delta\\ \Omega_{{\rm F}}^{{\rm eff}}/2&\xi_{k}-\lambda k&\Delta&0\\ 0&\Delta&-\xi_{k}+\lambda k&-\Omega_{{\rm F}}^{{\rm eff}}/2\\ -\Delta&0&-\Omega_{{\rm F}}^{{\rm eff}}/2&-\xi_{k}-\lambda k\end{array}\right],$$

$${\mathbf{P}}_F\left(\mu ,\theta ,\Delta \right)=\frac{1}{2}\sum _k(E_{k1}{\alpha }_{k1}^{†}{\alpha }_{k1}+E_{k2}{\alpha }_{k2}^{†}{\alpha }_{k2}$$

$$+E_{k3}{\alpha }_{k3}{\alpha }_{k3}^{†}+E_{k4}{\alpha }_{k4}{\alpha }_{k4}^{†})+\sum _k{\xi }_k-\frac{L{\Delta }^2}{g^F},$$

$$E_{k1,2}={\left({\xi }_k^2+{\eta }_k+{\Delta }^2\pm \sqrt{4{\eta }_k{\xi }_k^2+{\left({\Omega }_{\mathrm{F}}^{\mathrm{eff}}\right)}^2{\Delta }^2}\right)}^{\frac{1}{2}},$$

$${\mathbf{P}}_F\left(\mu ,\theta ,\Delta \right)=-\frac{1}{2}\sum _k\left(E_{k1}+E_{k2}\right)+\sum _k{\xi }_k-\frac{L{\Delta }^2}{g^F},$$

$$e^{iZ}=\stackrel{[}{a}=-b]b\prod \left[w_1^{\ast }\left(k_a\right)\cdot w_1\left(k_{a+1}\right)\right]\left[w_2^{\ast }\left(k_a\right)\cdot w_2\left(k_{a+1}\right)\right],$$

Full text

Introduction — In recent years, spin-orbit (SO) coupling in cold atoms SOCBosonExp2 ; SOC2 ; SOC1 ; SOC5 has received tremendous attention. Experimental realization of SO coupled bosons SOCBosonExp3 ; SOCBosonExp4 ; SOCBosonExp5 ; SOCBosonExp55 ; SOCBosonExp6 and fermions SOCFermiExp2 ; SOCFermiExp3 ; SOCFermiExp4 ; SOCFermiExp5 ; SOCFermiExp6 ; SOCFermiExp7 has been reported around the world. The interest in such systems is mainly due to the exotic phases induced by SO coupling in quantum gases. For example, SO coupled Bose-Einstein condensates (BEC) can host a stripe phase featuring spatially modulated density profiles SOCBosonTheory1 ; SOCBosonTheory2 ; SOCBosonTheory3 ; SOCBosonTheory4 ; SOCBosonTheory5 ; BFmixSOC1 ; BFmixSOC2 , which can be regarded as a supersolid SOCBosonExp6 ; liao ; whereas SO coupled attractive Fermi gas can become a topological superfluid, supporting Majorana edge states SOCFermionTopoBand1 ; SOCFermionTopoBand2 ; SOCFermionTopoBand3 ; SOCFermionTopoBand4 ; SOCFermionTopoBand5 ; SOCFermionTopoBand6 ; SOCFermionTopoBand7 ; SOCFermionMajorana1 ; SOCFermionMajorana2 ; SOCFermionMajorana3 ; SOCFermionMajorana4 ; SOCFermionMajorana5 . However, the spatial period of the stripe phase is on the order of the optical wavelength, making its direct observation extremely challenging, although indirect evidences for the stripe phase have been reported in two seminal experiments SOCBosonExp55 ; SOCBosonExp6 . Another serious experimental problem for realizing SO coupling in quantum gas concerns the heating due to the Raman beams. The Raman-induced heating is particularly severe for atomic species with small fine-structure splitting heating . Wei and Mueller carefully analyzed the heating problems for all alkali-metal atoms heating . According to their analysis, 40K and 6Li — the two most commonly used fermionic species in cold atom experiments — suffer greatly from such heating. This could explain why an SO coupled fermionic superfluid, despite its tremendous theoretical interest, has yet to be realized in experiment.

Here we consider a mixture of bosonic and fermionic superfluids, each of which is a spin-1/2 system. In addition to the density-density interactions, there exists an inter-species spin-exchange interaction. We assume that the condensate is subjected to the Raman-induced SO coupling, whereas the Fermi gas is not coupled by the Raman beams and hence is immune from the Raman-induced heating. The key observation of this Letter is that, through the spin-exchange interaction, the Fermi gas experiences a significant effective SO coupling. The interplay between the bosons and fermions leads to a variety of interesting quantum states, including a stripe phase in condensate with the spatial period much larger than the optical wavelength, and various topological phases for fermions. Note that a Bose-Bose spinor mixture has been realized in a recent experiment, and the associated effects of spin-exchange interaction were observed DJWang2 and theoretically analyzed BBmix . In addition, several groups have created Bose-Fermi superfluid mixtures with scalar condensates BFmixSuperfluid1 ; BFmixSuperfluid2 ; BFmixSuperfluid3 ; BFmixSuperfluid4 .

Hamiltonian — The total Hamiltonian of the system takes the form (we take $\hbar=1$)

[TABLE]

where $\Psi_{B}\left(x\right)=\left[\psi_{B\uparrow}\left(x\right),\psi_{B\downarrow}\left(x\right)\right]^{\mathrm{T}}$ represents the mean-field wave function of the BEC, and $\Psi_{F}\left(x\right)=\left[\psi_{F\uparrow}\left(x\right),\psi_{F\downarrow}\left(x\right)\right]^{\mathrm{T}}$ denotes the field operator of the Fermi gas. Both species have two internal spin states, which are labelled as $\uparrow$ and $\downarrow$. The single-particle Hamiltonians $h_{B}$ and $h_{F}$ are given by

[TABLE]

with $\Omega_{B}$ the Raman coupling strength, $k_{r}$ the Raman recoil momentum, and $\sigma^{x}_{B,F}$ and $\sigma^{z}_{B,F}$ the Pauli matrices. We set the two-photon detuning $\delta_{B}=\delta_{F}=0$ in our discussion. For simplicity, we assume a quasi-one dimensional system with strong transverse confinement. The key result that the fermions experience an effective SO coupling is insensitive to the dimensionality.

The last three terms in Eq. (1) describe three types of two-body interactions, where the Bose-Bose interactions read as

[TABLE]

with $\rho_{B\sigma}=\left|\psi_{B\sigma}\left(x\right)\right|^{2}$ the spin-$\sigma$ density of the BEC, the Fermi-Fermi interaction takes the form

[TABLE]

and the Bose-Fermi interactions are given by

[TABLE]

where $\rho_{B}=\rho_{B\uparrow}+\rho_{B\downarrow}$ is the density of bosons, and $\hat{\rho}_{F}=\psi_{F\uparrow}^{\dagger}\psi_{F\uparrow}+\psi_{F\downarrow}^{\dagger}\psi_{F\downarrow}$ is the density operator for fermions. Here we have assumed that the inter-species density-density interactions are spin-independent, with a single interaction strength $\gamma$, to avoid the proliferation of parameters. The last term in Eq. (6) describes the inter-species spin-exchange interaction characterized by the strength $\beta$. We take $L$ to be the length of the system with periodic boundary condition. The number of bosons and fermions are $N_{B,F}$, with the corresponding average densities $n_{B,F}=N_{B,F}/L$, respectively.

Non-interacting fermions — Let us first consider the case with non-interacting fermions, i.e., $g^{F}=0$. Previous studies of SO coupled BEC have shown that, in the absence of the fermions, the mean-field wave function of the condensate $\Psi_{B}$ can be accurately described by the following ansatz:

[TABLE]

where $k_{B}$, $\theta$, $C_{1}$ and $C_{2}$ are variational parameters. We can restrict $k_{B}\geq 0$ and $\theta\in\left[0,\pi\right]$ without loss of generality, and restrict $C_{1}$ and $C_{2}$ to be real positive numbers, with normalization condition $C_{1}^{2}+C_{2}^{2}=1$, as the relative phase between them will not affect the total energy. Based on the values of the parameters, three phases of the SO coupled BEC can be identified: the stripe phase (ST) with $B\equiv C_{1}C_{2}\neq 0$ and $k_{B}\neq 0$ where the condensate density profile shows the stripe pattern; the plane-wave phase (PW) with $B=0$ and $k_{B}\neq 0$ where the BEC condenses into a plane-wave state with finite spin polarization; and the zero-momentum phase (ZM) with $B=0$ and $k_{B}=0$ where the BEC features a smooth density profile with zero spin polarization. Given the variational ansatz (7), the BEC energy functional $\mathbf{E}_{B}\left(k_{B},\theta,B\right)$, corresponding to $\int dx\Psi_{B}^{*}h_{B}\Psi_{B}+\mathcal{G}_{B}$, is given by

[TABLE]

where we have defined $F(B)=\left(2G_{1}+4G_{2}\right)B^{2}-G_{2}$ and $G_{1,2}=n_{B}\left(g^{B}\pm g_{\uparrow\downarrow}^{B}\right)/2$.

The interplay between the condensate and the fermions is reflected in the $\mathcal{G}_{BF}$ term in Eq. (6). We include its effect in an effective fermionic single-particle Hamiltonian $h_{F}^{\rm eff}$ defined as

[TABLE]

Since Eq. (7) is quite general, we assume that the condensate wave function in the presence fermions can still be faithfully represented by Eq. (7). It follows that

[TABLE]

where

[TABLE]

The form of $h_{F}^{\rm eff}$ in Eq. (10) clearly shows that there is an effective SO coupling in the Fermi gas, which emerges from its interaction with the condensate. Since the two species influence each other, the SO coupling in both components possesses a dynamic nature. Dynamic synthetic gauge field has recently received much attention dsoc1 ; dsoc2 ; dsoc3 ; dsoc4 . The spinor mixture system thus provides another platform where dynamic SO coupling emerges naturally.

Diagonalizing $h_{F}^{\rm eff}$ gives a set of fermionic single-particle states. Then the total energy of fermions $\mathbf{E}_{F}\left(k_{B},\theta,B\right)$ is obtained by summing up the lowest $N_{F}$ eigenenergies of $h_{F}^{\rm eff}$. The ground state of the mixture is then obtained by minimizing the total energy functional $\mathbf{E}_{B}\left(k_{B},\theta,B\right)+\mathbf{E}_{F}\left(k_{B},\theta,B\right)$ with respect to the variational parameters. In our result, the final values of $\theta$ and $k_{B}$ roughly keep the relation $\cos(2\theta)\approx k_{B}/k_{r}$.

This procedure allows us to present the phase diagram of condensate in the $\Omega_{B}$-$\beta$ parameter space as shown in Fig. 1(a), where we take $n_{B}=10n_{F}$ and $N_{F}=2000$. To isolate the effect of the inter-species interactions, we take the density-density interaction strength $\beta=0$. In the absence of fermions, the condensate only possesses two phases, PW and ZM, for $g^{B}_{\uparrow\downarrow}>g^{B}$. The transition between them occurs around $\Omega_{B}=4k_{r}^{2}/(2m_{B})$. A notable feature of Fig 1(a) is that the region with large $\beta$ is dominated by the ST phase. This feature is clearly induced by the fermions. Specifically, the spin-exchange interaction induces an attractive interaction between the two spin components of the condensate, leading to a reduced effective $g^{B}_{\uparrow\downarrow}$, which favors the ST phase. The background color in Fig. 1(a) displays the value of $k_{B}$. In the ST phase, the condensate density profile is given by

[TABLE]

with a density modulation whose spatial period is determined by $1/k_{B}$. One can see that, for a given $\beta$, $k_{B}$ decreases as $\Omega_{B}$ increases. Figure 1(b) and (c) show two condensate density profiles corresponding to the black dot and yellow solid star in (a), respectively. For realistic parameters, the ST phase can possess a spatial period of several microns and a large modulation depth. Such a state can be readily observed using in situ imaging with today’s technology. The density profile for the Fermi gas, corresponding to the yellow solid star, is shown in (d). The two density profiles in (c) and (d) exhibit in-phase modulations.

Let us now turn to a more in-depth discussion of the properties of fermions in the mixture. When the condensate is in the PW or the ZM phase, we have $B=0$, and the effective single-particle Hamiltonian for fermions $h_{F}^{\rm eff}$ in Eq. (10) is reduced to (after neglecting a term proportional to the constant $n_{B}$)

[TABLE]

which has the same form as the Hamiltonian of an SO coupled Fermi gas, only that here the SO coupling is not due directly to the Raman lasers, but to the inter-species spin-exchange interaction with an effective Raman coupling strength given by

[TABLE]

When the condensate is in the ST phase, we have $B=1/2$, both $V$ and $M$ in Eq. (10) exhibit spatial modulations, originated from the density modulation of the condensate. The $V$-term, arising from the inter-species density-density interaction, serves as a lattice potential for the fermions, while the $M$-term, from the spin-exchange interaction, can be regarded as a periodic Raman coupling for the two spin components of the fermions. This situation is analogous to the optical Raman lattice proposed by Liu et al. SOCFermionMajorana1 ; Ramanlattice , and realized in recent experiments RLexp1 ; SOCFermionTopoBand1 . In the Raman lattice setup, the atom experiences an optical lattice potential and a periodic Raman coupling, both originated from the same laser beams. It is shown that the system parameters can be adjusted and induce topological phase transitions. Drawing from this analogy, we also expect topological phases in our system. Figure 2 displays the lowest four energy bands $E_{s}$ of the effective fermionic single-particle Hamiltonian $h_{F}^{\rm eff}$ in Eq. (10), when the condensate is in the ST phase. In all the plots in Fig. 2, we fix the value of $\beta$. Figure 2(a) is a reference plot where $\Omega_{B}=0$, hence there is no SO coupling in the system. Here different bands cross each other. The remaining three plots correspond to the same finite value of $\Omega_{B}$, with varying $\gamma$. In these cases, gaps open up at band crossing points in (a). The color of each band represents the spin polarization $P=\langle\sigma_{F}^{z}\rangle$, which can be seen to be momentum-dependent — a manifestation of the SO coupling. The values of $Z_{j}$, indicated in (b) and (d), are the Zak phase for each band, defined as Zak

[TABLE]

where $w_{j}\left(k_{a}\right)$ is the eigenstate of band $j$ and discretized momentum $k_{a}$, restricted in the first Brillouin zone in the range $k_{a}\in[-k_{B},k_{B})$, with the additional constraint $w_{j}\left(k_{d+1}\right)=w_{j}\left(k_{-d}\right)$ to form a loop in the calculation of the Zak phase. At a critical value of $\gamma$ shown in Fig. 2(c), the lowest two bands crosses each other. When the band reopens at a larger value of $\gamma$, the Zak phase of some of the bands changes its value. Thus the closing and the reopening of the band gap signals a topological transition. Note that Zak phase in topological Bloch bands has been measured in recent cold atom experiments zakexp .

Interacting fermions — Now let us turn to the situation where the fermions are self-interacting with an attractive $s$-wave interaction strength $g^{F}=-6k_{F}/\left(\pi m_{F}\right)$ in Eq. (5), which can lead to superfluid pairing. Including Fermi-Fermi interaction greatly complicates the physics in the ST phase STint . To keep things relatively simple, we take a large boson density $n_{B}=500n_{F}$, such that in the parameter space we will explore, the bosons are nearly unaffected by the fermions and remain in either the PW or the ZM phase. Under this situation, the effective fermionic single-particle Hamiltonian is given by $h_{\rm F,PW}^{\rm eff}$ in Eq. (14). We thus have a system of attractive Fermi gas subjected to SO coupling with an effective Raman coupling strength $\Omega_{F}^{\rm eff}$ defined in Eq. (15). The corresponding fermionic system has been studied before SOCFermionTopoBand6 and is known to support topological superfluid phase. This can be intuitively understood as follows. The SO coupling mixes spin singlet and triplet pairings. The Raman term, which can be regarded as an effective Zeeman field, tends to weaken the singlet pairing. For sufficiently large $\Omega_{F}^{\rm eff}$, the singlet pairing is suppressed, and the Fermi gas becomes a topological superfluid with effective $p$-wave pairing. This picture is indeed confirmed by our calculation.

Our calculation proceeds as follows. The bosonic contribution to the total energy functional $\mathbf{E}_{B}\left(k_{B},\theta,B\right)$ still takes the form of Eq. (8), only that the parameter $B$ vanishes for the PW and the ZM phase. For the fermionic part, the thermodynamic grand potential can be written as

[TABLE]

where $\mu$ is the chemical potential, and $h_{{\rm F,PW}}^{{\rm eff}}$ and $\mathcal{\mathcal{G}}_{F}$ are defined by Eqs. (14) and (5), respectively. Note that $h_{{\rm F,PW}}^{{\rm eff}}$ is a function of the variational parameter $\theta$. In our treatment of the fermionic part, we follow the standard mean-field approach introduced for the single-species interacting fermions as reported in, e.g., Refs SOCFermionTopoBand6 ; SOCFermionMajorana4 . In the mean-field approximation, the Fermi-Fermi interaction term becomes

[TABLE]

where the superfluid order parameter is defined as

[TABLE]

In momentum space, the grand potential can be re-written as

[TABLE]

where C_{k}^{\dagger}=\left[\begin{array}[]{cccc}c_{k\uparrow}^{\dagger}&c_{k\downarrow}^{\dagger}&c_{-k\uparrow}&c_{-k\downarrow}\end{array}\right] and

[TABLE]

with $\xi_{k}=k^{2}/\left(2m_{F}\right)-\mu$ and $\lambda=-k_{r}/m_{F}$. Diagonalizing the matrix $M_{k}$, we can further transform the grand potential to the following form:

[TABLE]

where $\alpha_{k1}$, $\alpha_{k2}$, $\alpha_{k3}$, and $\alpha_{k4}$ are quasi-particle elementary excitation operators with the symmetry $E_{k4}=-E_{-k1}$ and $E_{k3}=-E_{-k2}$. The two positive excitation branches are given by

[TABLE]

with $\eta_{k}=\left(k_{r}k/m_{F}\right)^{2}+\left(\Omega_{{\rm F}}^{{\rm eff}}\right)^{2}/4$. The ground state of the fermions is considered to be the quasi-particle vacuum, with the corresponding ground-state grand potential given by

[TABLE]

where the anti-commutation relations of $\alpha_{k1}$, $\alpha_{k2}$, $\alpha_{k3}$, and $\alpha_{k4}$ have been considered. Note that $\mathbf{P}_{F}\left(\mu,\theta,\Delta\right)$ is a functional of three undetermined variational parameters $\mu$, $\theta$, and $\Delta$.

The ground state of the whole mixture is obtained through the minimization of $\mathbf{E}_{B}\left(k_{B},\theta,0\right)+\mathbf{P}_{F}\left(\mu,\theta,\Delta\right)$ with respect to the variational parameters $k_{B}$, $\theta$, and $\Delta$, where the constraint $N_{F}=-\partial\mathbf{P}_{F}\left(\mu,\theta,\Delta\right)/\partial\mu$ is imposed to fix the number of fermions. Actually, the minimization of $\mathbf{E}_{B}\left(k_{B},\theta,0\right)$ with respect to $k_{B}$ leads to the rigorous relation $\cos\left(2\theta\right)=k_{B}/k_{r}$, and hence we only need to deal with $\theta$ and $\Delta$ in the numerical minimization. The converged results of $k_{B}$, $\theta$, $\Delta$, and $\mu$ are obtained in the thermodynamic limit with $N_{B},N_{F},L\rightarrow\infty$ while keeping $n_{B}=N_{B}/L$ and $n_{F}=N_{F}/L$ finite.

Our results are summarized as follows. Figure 3(a1) represents the zero temperature phase diagram of the Fermi gas in the $\Omega_{B}$-$\beta$ space. It shows three phases: the non-topological superfluid (SF), the topological superfluid (TSF), and the normal phase. The first two phases feature finite superfluid order parameter $\Delta$, whereas $\Delta$ vanishes in the normal phase, as shown in Fig. 3(a4). The quasi-particle excitation gap $E_{g}$ is finite in the SF and the TSP phases, except at the boundary of these two phases, as shown in Fig. 3(a2), where $E_{g}$ vanishes as expected for topological phase transition. Several examples of the quasi-particle excitation spectra $E_{k}$ at various phases are displayed in Fig. 3(b1)$\sim$(b4).

The topological phase transition can be further confirmed by the winding number $Z$, which is defined through a loop connecting the two positive excitation branches at infinitely large $k$, and can be calculated as SOCFermionMajorana4

[TABLE]

where $w_{1}\left(k_{a}\right)$ and $w_{2}\left(k_{a}\right)$ are the quasiparticle eigenstates corresponding to the two positive excitation branches $E_{k1}$ and $E_{k2}$ in Eq. (22), with the quasi-momentum defined as $k_{a}=2\pi a/L$ with the integer $a\in\left[-b,b\right]$ where $b$ is a sufficiently large numerical cut-off. We define $w_{1}\left(k_{b+1}\right)=w_{2}\left(k_{-b}\right)$ and $w_{2}\left(k_{b+1}\right)=w_{1}\left(k_{-b}\right)$ to connect the two positive excitation branches and form the loop in calculating the winding number. Note that here $k_{a}$ in (24) is no longer restricted in the first Brillouin zone. As shown in Fig. 3(a3), $Z$ jumps from $\pi$ to 0 when entering from SF to TSF. Finally, through combining the result of $k_{B}$ in Fig. 3(a5), the definition of $\Omega_{F}^{\rm eff}$ in Eq. (15), and the relation $\cos(2\theta)=k_{B}/k_{r}$, we can see that the TSF phase does correspond to the larger $\Omega_{F}^{\rm eff}$, and hence our result agrees with our previous picture.

Summary — In summary, we have investigated a system of Bose-Fermi spinor mixtures. The bosons form a condensate that subjected to the Raman-induced SO coupling, while the fermions are not coupled to the Raman lasers, but interact with the bosons via the density-density and/or spin-exchange interaction. We show that the spin-exchange interaction makes the fermions experiencing an effective SO coupling, without suffering Raman-induced heating. This could pave a new way towards the first realization of SO coupled fermionic superfluids, which can be made to be topological with a proper choice of parameters. The interplay between the bosons and fermions also has an interesting effect on the former: the Bose-Fermi interaction favors the condensate to be in the ST phase, with an interaction-dependent spatial modulation period. With realistic parameters, the spatial modulation period can be as large as several microns, making the ST phase readily observable with the in situ imaging technique. This provides a significant advantage in both the realization and the observation of the ST phase. The phenomena described above arise due to the emergent and dynamic nature of the SO coupling in spinor mixtures.

Our proposal does not require any new experimental techniques beyond those that have already been demonstrated in the lab. In particular, both the Bose-Bose spinor mixtures DJWang2 and the Bose-Fermi superfluid mixtures BFmixSuperfluid1 ; BFmixSuperfluid2 ; BFmixSuperfluid3 ; BFmixSuperfluid4 have been realized. The density-density interaction strengths have been routinely tuned via Feshbach resonance Feshbach2 ; Feshbach3 . Recent works have also shown that the spin-exchange interaction can be tuned to some extent SpinExchInt1 ; SpinExchInt2 ; SpinExchInt3 ; SpinExchInt4 ; SpinExchInt5 . We therefore expect that our proposal will inspire more works, both theoretical and experimental, on SO coupled quantum gas spinor mixture, providing a new and unique platform to explore the physics of SO coupling.

We acknowledge discussions with Xiong-Jun Liu, Randall G. Hulet, and Peng Zhang. HP is supported by the US NSF and the Welch Foundation (Grant No. C-1669), LC by the NSF of China (Grant No. 11804205), HH and XJL by the Australian Research Council (Grant No. DP170104008).