Orbital Feshbach resonance of Fermi gases in an optical lattice

In this paper, we study the orbital Feshbach resonance of ultracold two-electron 173Yb Fermi gas in a one-dimensional optical lattice. We consider a two-band model under the assumptions of a local-density approximation for the optical lattice potential and a mean-field approximation for the intraband Cooper pairings. We get three crucially important properties for the system, the superfluid-Landau–Fermi-liquid (LFL) crossover corresponding to the tunneling energy, the particle condensation in the momentum space, and the superfluid-LFL phase transition corresponding to the temperature. We explain how to realize and manipulate these properties.


Introduction
The theoretical study of Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover dates back to the seminal works of Eagles and Leggett, where they pointed out that the BCS wave function can be extended to describe the physics beyond the weak-coupling limit [1,2]. Despite being a mean-field theory, the approach paints a qualitatively correct picture throughout the BCS-BEC crossover at zero temperature. Here the s-wave scattering length between different fermions can become infinitely large at Feshbach resonance. For alkali atoms with spin S=1/2, the relative energy of spin singlet and triplet can be controlled by using the Zeeman field to reach a magnetic Feshbach resonance (MFR). On the other hand, there is no MFR in alkali-earth atoms due to the absence of total electron spin in their ground states [3,4].
In alkali atoms, such as 6 Li and 40 K, the spin singlet and triplet molecular scattering states can be tuned to resonance by an applied external magnetic field due to different magnetic moments for MFR [5,6]. However, in alkali-earth atoms, such as 173 Yb, there is no MFR between the excited-state 3 P 0 and ground state 1 S 0 [4,7]. For such atoms, the orbital Feshbach resonance (OFR) due to the spin-exchange interactions and the nuclear Landé g factors between different orbital states get the key importance.
Recently, theoretical and experimental works on two-band descriptions of an OFR have made significant progress [3,[7][8][9]. For an orbital resonance, the two-band theory describes scatterings of the open and closed channels and their mutual interaction. On the other hand, a single-channel theory is only related to the description of the open channel. In the limiting case of low atomic density, results of the single-channel theory match with the two-band theory [10]. In an interesting study [9], for the trapped 173 Yb Fermi gas, the interplay between the pairbreaking and the thermal-broadening effects was examined by tuning temperature of the Fermi gas.
For Fermi gases in optical lattices, Jördens et al [11] found that the interaction strength can be directly controlled to compare the Mott insulating regime and the non-interacting regime without changing the tunnel-coupling or confinement. Similarly, a direct measure of the compressibility of quantum gas in an optical lattice may be realized by using in situ imaging and independent control of external confinement and lattice depth [12]. There are several interesting studies related to the ultracold atoms in the optical lattice which is a defect free set up that allows independent control of various parameters of the system [13][14][15]. Here, the high controllability over the optical lattice parameters permits one to use the internal atomic degrees of freedom for extension of a onedimensional (1D) optical lattice into a synthetically yielded two-dimensional lattice [16][17][18][19][20].
In this paper, we consider two nuclear spin states denoted by ñ | (m F =1/2) and ñ | (m F =−1/2) for OFR, e.g., the two nuclear spin states in 173 Yb (I=5/2) with m , , ; 2 ñ =  ñ - ñ | (| | ) and the closed channel c g e e g ; ; , introducing the coupling between the orbit and spin degree of freedom [4,[21][22][23]. We consider a two-band model under the assumptions of a local-density approximation for the optical lattice potential and a mean-field approximation for the intraband Cooper pairings. We find that OFR detuning gradually decrease the critical tunneling energy for superfluid-normal phase transition at zero temperature and there is a narrow superfluid phase region between the two normal phases for a large OFR detuning. We also find that the critical superfluid-normal transition temperature may be reduced even to zero for increasing the tunneling energy of the 1D optical lattice.
Rest of the paper is organized as follows. In section 2, we give the Hamiltonian of OFR in an optical lattice and deduce the thermodynamic potential in momentum space by obtaining the self-consistent equations. In section 3, we discuss obtained results of OFR by tuning different controlling parameters of the system respectively. Finally, section 4 is devoted to the essential conclusions of the work.

Model and dynamics
We consider a Fermi gas in 1D optical lattice potential ) generated by a standing wave field at the 'magic' wavelength λ L =759 nm, to ensure the lattice dispersions are the same for all states in two channels, as shown in figure 1. The potential strength V 0 is proportional to the laser intensity and corresponds to the maximum depth of the lattice potential. The Hamiltonian describing Fermi gases across an OFR is exactly analogous to that of two-band s-wave superconductors with contact interactions in an optical lattice [8,11,12,21] and we can get the orbital Feshbach two-band Hubbard Hamiltonian by using the single-particle Wannier basis as , , To solve the above Hamiltonian, it is convenient to work in the momentum space. For this goal, the equation (1) can be transformed to an expression in momentum space by using Fourier transformation. We introduce an intraband order g¢ being the thermal average under the mean-field pairing approximation, and the resulting Hamiltonian now reads The operators c ksg † create a single-particle with the spin σ in channel γ, momentum k and dispersion ξ kγ =ε k −μ γ , where k m 2  minimizing the ground state thermodynamic potential in the grand canonical ensemble. The self-consistent parameters are given by | | is the energy of the quasiparticle excitations in the γ channel, T represents temperature of the Fermi gas, and n is the density of atoms. By solving the self-consistent equations, we find that there are two solutions, i.e., an in-phase solution for both Δ 2 and Δ 1 having the same signs and an out-of-phase solution for both Δ 2 and Δ 1 having different signs. However, the in-phase solution is trivial, which has nothing to do with the BCS-BEC crossover associated with the OFR [24]. We will focus on the discussion of the out-of-phase solution of the two pair potentials, which is, in fact, an excited-state in the landscape of the thermodynamic potential. The excited-state (out-of-phase) solution is predominantly characterized by the OFR in a 173 Yb Fermi gas [8].

Results and discussions
In this section, we want to examine how do the basic natures of the two-channel system, such as order parameter (Δ γ ), density of atoms (n γ ) and momentum distribution (n γk ) of the ultracold Fermi gas, vary corresponding to the dimensionless parameters t/E F (the tunneling energy), T/E F (the temperature) and δ/E F (the OFR or related to the magnetic field). That is, for convenience, we use E k m 2  The order parameter Δ γ , and density fraction n γ /n can be tuned by the tunneling energy t according to equations (5)-(7). In  figures 2(a) and (b), we plot the order parameter Δ γ /E F and the particle density n γ /n versus the tunneling energy t/E F for two channels for different values of δ, respectively. One can see that, for relative small tunneling energy t and small δ the description of the two channels model is important, because in this case two channels have different properties of superfluidity and particle densities. Figure 2(a) shows that for small t and δ it is easy to form Cooper pairs and the system exhibits the superfluidity. When t is large enough (for any δ), Δ γ (for γ=1, 2) tend to zero. This means Cooper pairs disappear and the system goes to normal phase. This property can be referred to as the superfluid-normal phase crossover corresponding to the tunneling energy. For a deep lattice potential (V E 1 0 F  ), the hopping between neighboring sites is forbidden and atoms are trapped in separated lattice sites, or say in the harmonic potential [9]. For a nonzero tunneling energy t, atoms can hop between neighboring sites. The hopping can prevent from the formation of pairs on all sites and then with t becoming large, the system carries out a crossover from superfluid to normal phase. It is worth noticing that δ also influence the superfluidnormal crossover. When δ increases, the superfluid-normal crossover shifts toward a smaller t. An interesting phenomenon is that unlike for a small δ case, at a large δ, for example δ=10E F , the order parameters of two channels have an individually increased peak. Figure 2(b) shows the variation of n γ /n (particles in two channels) versus t/E F (the tunneling energy) for different values of δ. The relation with δ means that the particle densities can be adjusted by the magnetic field according to the definition of δ. One can see that, in the region t E 1.5 F < , the particle density in the open channel is usually larger than that in the closed channel. If the magnetic field intensity (as well as δ) is increased, the particle density in the open channel increases accordingly, even up to all occupation in the open channel when the magnetic field is large enough. There is a critical value, δ th /E F =3.14. Above this value (for large δ), the particle density for the closed channel may rapidly tend to zero (the closed channel becomes empty), then all particles are located in the open channel. However, when δ/E F <3.14, the system usually has n 2 <n 1 , but it normally does not tend to zero. One can see from the figure that all curves vary smoothly, i.e. no sudden change happens. Then no phase transition happens corresponding to the tunneling energy for the system.
We may explain the properties of the system how does it look like in this case. According to the definition of E F (Fermi energy), for a non-interacting single-channel Fermi gas at zero temperature, the upper closed channel is completely empty for the energy difference δ/2>E F . Now in this twochannel system, for all possible δ with small t, particles are scattered in two channels, this is because of the coupling between two channels. However, for the case E 2 F d  , the two-channel coupling is not strong enough to overcome the detuning barrier of the Fermi gas. As a consequence, particles in the closed channel vanish and only concentrate in the open channel, which leads to the two-channel pairings collapsing as well (Δ γ =0). One may notice that, for a small OFR detuning (δ/E F <3.14) (for example, δ/E F =1), the densities n 1 and n 2 behavior oppositely. First of all, we have to emphasize that n 1 +n 2 =n, the total particles are conservative, so when n 1 increases, n 2 must decrease, and vice versa. The variation of n is connected with Δ γ , the chemical potentials μ γ , and so on. These connections carry out n having a nonlinear relation with the tunneling energy t, which is especially connected with the formation and variation of Cooper pairs.

A condensation in the momentum space
The existences of the recoil energy E k m 2 in optical lattice and the tunneling effect may influence the momentum distributions of particles. So we can investigate the relation of particle numbers with the relation of momentum, and define the particle density of the momentum distributions of the ultracold Fermi gas as n γk , and n n n dk k ò = g g is the same as given in equation (7). In figure 3, we plot the momentum density distributions of two channels with the variation of k/2k L under the conditions of t=0 and E 0.45 F . One can see that in the case t=0 (no tunneling energy), both n γ1 and n γ2 vary continuously and smoothly. However, for t=0.45 the variation of curves are quite different. They show that there is another peak correspondingly. This is a very interesting property. To examine why this peak happens, in the inset we plot a figure of ξ kγ /E F versus k k 2 L . One can find that two peaks of n γk are exactly matching the positions of the minima of ξ kγ . While ξ kγ is physically related with the energy of Fermi gas, it is this relation leads to the peak of the distribution. The first peak is well understood because for most particles of the ultracold Fermi gas their momenta must be very small, so there are a lot of particles locate in a small momentum region. Then for the minimum ξ kγ value, there is another peak, the accumulation of particles at this position are referred to as the particle condensation in the momentum space. This is an important property of the system. Through calculation we found that the dispersion exists another minimum point only if the tunneling energy t/E F >0.155. Otherwise, the peaks of the momentum distribution will disappear. This is also the condition of the particle condensation in the momentum space.

The phase transitions corresponding to the temperature
The characteristic with temperature T of the Fermi gas is another important nature. In figures 4(a) and (b), we plot the order parameters and densities of two channels versus with the temperature T. How these parameters vary with the temperature is obvious and intuitionistic. It is not necessary to describe again in more detail. However, we have to emphasize a very important physics meaning of curves, i.e. the phase transitions corresponding to the temperature of the system. Let us see figure 4(a) first, it shows that both channels are in the superfluid states at low temperature. Up to the critical temperature T c which is dependent on the tunneling energy, two channels of the system undergo a transition from the superfluid states to the normal state (Δ γ =0), simultaneously. This is the analog of the typical phase transition in condensed matter physics. So this result shows the superfluidnormal phase transition of a two-channel ultracold Fermi gas corresponding to the temperature in the optical lattice.
This phase transition is also reflected in the nature of the particle densities of two channels as shown in figure 4(b). One can see that two-channel densities vary with the increase of the temperature, up to the critical temperature, the densities of two channels undergo a transition which is shown by the variation of the derivative of the densities. The discontinuity of the derivative of the densities expresses a phase transition from an orderly superfluid phase to a disordered normal phase, which corresponds the Landau-Fermi-liquid (LFL). Over the phase transition point, the 'pairing' order in the superfluid is broken down. In the superfluid phase (T<T c ), the particle density bias of two channels (n 1 −n 2 ) increases with temperature. It illustrates that as the temperature rises, more and more atoms accumulate to the open channel of the lower energy, and meanwhile the particles in the upper closed channel are exhaled. In contrast, for the normal phase (T>T c ), the particle density bias of two channels decreases with the temperature.
From figures 4(a) and (b), one can see that the phase transition here is actually related to the tunneling energy of the system. So, we are interested in how this relation is. For this purpose, we plot figure 4(c), which is the relation of the critical temperature T c versus the tunneling energy t. It is worth noticing that the critical superfluid-normal transition temperature is reduced with increasing the tunneling energy t. Actually for t E F  , Fermi atoms do not form the superfluid order at all, even in very low temperature. So the critical temperature tends to zero for a large value of t.

Additional remarks
Up to now, we have actually got all the basic results of the system. In this subsection, we plot figure 5 as additional remarks for detailed influences of OFR detuning on the behavior of the Fermi gas in the optical lattice. Figures 5(a) and (b) show the variation of the order parameters and the particle densities with the OFR detuning δ under the condition t E 0.45 F = . These two figures show explicitly how does the magnetic field control the evolutions of the order parameters and the particle densities. In (c) The critical temperature T c of the superfluid-LFL phase transition versus the tunneling energy t. All curves are plotted for the OFR detuning δ th /E F =3.14. figure 5(a), one can see also that the order parameters undergo a superfluid-normal crossover corresponding to δ. In figure 5(b), one can see that with the particles accumulate in the open channel, the particles exhale from the closed channel. Figure 5(c) shows the particle condensation in the momentum space as shown before. One can see that, when δ becomes large, the condensation becomes much more evident.

Conclusions
In summary, we have studied OFR of 173 Yb ultracold atoms in a 1D optical lattice. The investigation is based on the excited-state (out-of-phase) solution of a two-channel model. We found three fundamentally important properties: the superfluid-LFL crossover corresponding to the tunneling energy, the particle condensation in momentum space, and the superfluid-LFL phase transition corresponding to the temperature. All these properties can be manipulated by the OFR. For ultracold atoms, it is understood that the distribution corresponding to the momentum must be nonuniform. But a condensation corresponding to the momentum obtained here is unexpected. This condensation is especially evident when the OFR detuning δ is large as shown in figure 5(c). In this case, the condensation happens mainly in two areas in the open channel: k k 0 2 0.4 L   and 0.7  k k 2 1.1 L  . The reason for this momentum condensation relies on the property of the energy dispersion relation. A phase transition in any system is crucially important in physics. The superfluid-LFL phase transition corresponding to the temperature here can also be manipulated by the tunneling energy t and OFR detuning δ. It is found that the critical superfluid-normal transition temperature may be reduced even to zero with increasing values of the tunneling energy t. The superfluid phase disappears as the critical superfluid-normal transition temperature is T c ≈0 for the tunneling energy t E 1.7 F  . In general, for a given system, t may not be changed, so the OFR detuning δ is a very suitable parameter for controlling the critical temperature of the phase transition. The method developed in this paper can be extended to the analysis of other related problems in an optical lattice, such as the chiral edge currents of these topological edge states in synthetic ladders [17,18,[25][26][27][28].