Quantum Simulation of Competing Orders with Fermions in Quantum ...

Quantum Simulation of Competing Orders with Fermions in Quantum Optical Lattices Arturo Camacho-Guardian,1 Rosario Paredes,1 and Santiago F. Caballero-Ben´ıtez∗2 1

arXiv:1706.02347v1 [cond-mat.quant-gas] 7 Jun 2017

Instituto de F´ısica, Universidad Nacional Aut´ onoma de M´exico, Apartado Postal 20-364, M´exico D. F. 01000, Mexico. 2 ´ CONACYT-Instituto Nacional de Astrof´ısica, Optica y Electr´ onica, Calle Luis Enrique Erro No. 1, Sta. Mar´ıa Tonantzintla, Pue. CP 72840, M´exico (Dated: June 9, 2017)

∗

Ultracold Fermi atoms confined in optical lattices coupled to quantized modes of an optical cavity are an ideal scenario to engineer quantum simulators in the strongly interacting regime. The system has both short range and cavity induced long range interactions. We propose such a scheme to study the interplay of density order, antiferromagnetic states, and superfluidity in the Hubbard model in a high finesse optical cavity at T = 0. We demonstrate that those phases can be accessed by properly tuning the linear polarizer of an external pump beam via the cavity back-action effect, while modulating the system doping. This emulates typical escenarios of analog strongly correlated electronic systems.

Coupling ultracold quantum gases to high-finesse optical cavities is a novel scenario to explore many-body phases in the full quantum regime by exploiting the controllability of light-matter interaction [1, 2]. Major experimental breakthroughs have been achieved in the quantum limit of both light and matter. For instance, the transition to Dicke phase has been observed in a Bose-Einstein condensate coupled to cavity modes [3]. Experimentally, it has been achieved the emergence and control of supersolid phases where the backaction of the cavity generates a light-induced effective long-range interactions which competes with the short-range interatomic interactions [4–6]. On the theoretical side, recent studies have introduced settings where the cavity fields generate gauge-fields [7, 9, 10], artificial spin-orbit coupling [11], self-organized phases [12], topological phases [7, 13, 14], induced magnetic and density order using the measurement back action [8], dimerization [15], spin lattice systems[16] and quantum simulators based on global collective light-matter interactions [17, 18]. Ultracold Fermi gases loaded in optical lattices with long-range interactions using magnetic polar interactions and electric dipolar interactions are possible [19]. However, the temperatures needed to investigate the competition between the different orders and the fact that the interactions depend on the particular constituents pose some limitations. Additionally, extended Bose-Hubbard systems [20] have been achieved, Rydberg systems [21] have been proposed, while small systems with ions [22] or analogous superconducting circuits [23] and multi-mode opto-mechanics are also possible [24]. Ultracold gases in OL inside cavities allow to engineer the spatial structure of the many-body interactions, that beyond dipolar systems, is independent of the intrinsic nature of the interaction. The effective light-induced many-body inter-

∗ Electronic

address: [email protected]

FIG. 1: (Color online) Schematic representation of the model. A Fermi-Hubbard chain is placed inside a single mode optical cavity. A linearly polarized probe beam with incidence perpendicular to the cavity along the z axis, creates an electromagnetic field that couples light to the atomic modes. The disk shows the choice of polarization angle θ measured from the -x polarization axis. The light pumped into the system induces density wave order maximally for θ = 0 (bottom) while antiferromagnetic order is maximal for θ = π/2 (top).

action can be engineered by external means, controlling the properties of the light pumped into the system. Simulating quantum many-body long-range Hamiltonians with emergent quantum phases of matter complements current efforts to understand condensed matter phenomena [25]. In contrast to condensed matter systems, ultracold quantum gases can be measured with a single site resolution [26]. This allows to extract the density distribution and non-local correlation functions [27]. Using this information in experiments, it has been possible to produce and detect the properties of 1D Hubbard chains seeking for hidden antiferromagnetic correlations [28] with the aim of improving our knowledge on the phase diagram of the two dimensional Hubbard model [29–31]. In this Letter, we propose a route to study the emer-

2 gence of quantum phases of fermionic matter as a result of the competition between of short and long-range lightinduced interactions via cavity fields. The latter type of interactions is absent in standard (“classical”) optical lattices without cavities. For this purpose, we consider a mixture of Fermi atoms in two hyperfine states confined in an optical lattice in one dimension. The atomic chain lies in a single mode cavity and is illuminated by a coherent pump beam, see Fig1. We tune the polarization angle of the pump beam and show that this alllows to manipulate the emergence of different phases of quantum matter in the system. The system will support the formation of density wave (DW) insulators, antiferromagnets [32], pair-superfluid states (SFη ) [33] or pair density waves (PDW) [34, 35]. We show that by controlling the polarization of the pump beam, it is possible to manipulate which symmetries of the system break. Therefore, we can instigate the formation of the aforementioned phases at will, with full parametric control by external means using the pump beam. The ultracold fermions are loaded in an OL aligned with the cavity in the x direction and a coherent standing wave illuminates the optical cavity along the z axis. This creates a long-range interaction via off-resonant scattering into the optical resonator mode [2]. In the steady state of light, the effective state of quantum matter is given by the following atomic Hamiltonian [15, 18, 36], ˆ eff = H ˆ f + geff (D ˆ †D ˆ +D ˆ D ˆ† H (1) θ θ ), 2Ns θ θ P ˆ† ˆ f = −t P P where H ˆ i↑ n ˆ i,↓ , σ hi,ji (fiσ fjσ + H.c.) + U in is the standard Hubbard Hamiltonian with t the tunneling amplitude, U the interspecies on-site interaction strength and Ns the number of sites, this is the classical optical lattice contribution. The tunneling amplitude has a direct dependence on the lattice depth V0 of the OL [37]. For our study, we consider lattice depths where the single band approximation is valid, namely, V0 & 5ER , being ER the recoil energy. The effective light-induced structured infinite range interaction energy is described by the last term of the Hamiltonian, where geff is its effective coupling strength. This is the quantum optical lattice (QOL) contribution [2, 18] that depends on the cavity back-action of the system (see [38] for details). Thus, the Hamiltonian describing the system can be referred as a fermionic quantum optical lattice Hamiltonian (FQOL). The effective light-induced interaction strength geff = ∆c |˜ gp |2 |JD |2 Ns /(∆2c + κ2 ) ≈ |˜ gp |2 |JD |2 Ns /∆c ∝ V0 /∆c , where g˜p = gc gp ap,θ /∆a , is the two photon Rabi frequency with gc/p being the light-matter coupling coefficients, ap,θ the pump amplitude, ∆a the detuning between the pump frequency and the atomic resonance, and ∆c = ωp − ωc the detuning between the pump beam ωp and the cavity mode ωc . The pump-cavity detuning allows for the control of the sign of geff [4]. The constant JD ∼ 1, is the amplitude of the overlap integral between the cavity, pump and classical optical lattice on-site Wannier functions Ji,i [38]. Assuming well-

localized atoms (V0 ∼ 10ER ), we can neglect additional off-diagonal coupling contributions given by the inter-site overlap integrals. Off-diagonal coupling terms may lead to other exotic states of matter as shown in the bosonic analog [15, 18] and will be considered elsewhere. The system under study is in the limit where cavity back-action is dominant and measurement back-action is negligible ∆c  κ. The polarization of the probe beam controls the effective spatial and inter-species coupling in the QOL through X ˆθ = D Jj (ˆ nj,↑ eiθ + n ˆ j,↓ e−iθ ). (2) j

The polarization of light along −x axis in the effective ˆx = D ˆ θ=0 = Hamiltonian couples the density modes via D P J ρ ˆ , with ρ ˆ = n ˆ +ˆ n , while polarization in −y dij j↑ j↓ j j j rection couples the fermions through their magnetization ˆ y = −iD ˆ θ=π/2 = P Jj m D ˆ j , with m ˆj = n ˆ j↑ − n ˆ j↓ . For j geff < 0, the effective interaction induced by light creates a staggered field that breaks the translational symmetry generating a predominant DW phase for 0 ≤ θ < π/4, while inducing predominant antiferromagnetic order for π/4 < θ ≤ π/2. Indeed, at θ = π/4, maximum competition between orders takes place, as they both become favored in the same proportion. The coefficients Ji can be chosen arbitrarily by properly selecting the angles between the chain, the cavity and the probe beam. Here we consider the case where the cavity is placed along the chain and the probe beam is perpendicular to them (at 90◦ ), thus giving a spatial profile for the coupling coefficients Ji = (−1)i . The angle of the linear polarization of the probe beam, together with doping (number of holes) of the model allow to engineer phase paths in order to study the competition between several quantum phases. The atoms will self-organize, optimizing the energy with maximal light scattering geff < 0 and minimal for geff > 0[18]. In the following, we consider geff < 0 and U < 0, on-site attraction between species for simplicity, additional results will be reported elsewhere. In Fig. (2) we show that a variety of routes (trajectories) can be created. For example the route [Fig.(2) (a)] SFη → PDW → AFM → PDW → SFη can be designed, or a path [Fig.(2) (b)] SFη → AFM → DW → AFM → SFη can be made. This allows us to emulate the strongly correlated regime of fermionic quantum matter analogous to a prototypical scenario of a High-Tc superconductor [34]. In our system (analogous) the superfluid (superconducting), density (charge) wave and magnetic (spin) order parameters show the competition between the different quantum many-body phases. We show the behavior of the order parameters associated to each phase in Fig.(2).

3 TABLE I: Relation between parameters and quantum manybody phases (QP) QP/Parameter ODW Oη OM

SFη DW AFM PDW

FIG. 2: (Color online) Competition of quantum phases by tuning the angle θ for a given value of the doping. Order parameters ODW , OM and Oη as functions of the filling factor and the angle of polarization θ. In panel (b), at half-filling an AFM phase is produced with θ = π/2. Properly changing the angle of the linear polarized beam to θ = 3π/8, a PDW phase emerges doping the system by one or two holes/pairs. For values of doping far beyond of half-filling and θ = π/4, a SFη phase dominates. In panel (b), a DW insulator is produced at half-filling with θ = 3π/10. Doping by one pair/hole and changing the polarization angle to θ = 2π/5, the system is an AFM. A SFη phase can be accessed away from halffilling by doping further and increasing the polarization angle to θ = π/4. Parameters are: Ns = 10, t/|U | = 0.1, geff /|U | = −0.75 (a) and geff /|U | = −1 (b).

Those order parameters were obtained directly from the calculation of the ground state correlation functions, as well as, taking into consideration the following definitions, SM (q) =

1 X iq(j−l) e (hm ˆ jm ˆ l i − hm ˆ j ihm ˆ l i), (3) Ns j,l

1 X iq(j−l) SDW (q) = e (hˆ nj n ˆ l i − hˆ nj ihˆ nl i), Ns

(4)

j,l

(ρ2 )i,j = hˆ c†i,↑ cˆ†i,↓ cˆj,↓ cˆj,↑ i,

(5)

where SM/DW is the magnetic/density structure factor and ρ2 is two-body reduced density matrix. Magnetic order is characterized through the structure factor SM (q) being q the magnitude of a wavevector in the first Brillouin zone. We use the AFM parity as the ˆ y2 i/Ns2 . The order parameter OM = SM (q = π)/Ns = hD density wave order parameter is ODW = SDW (q = π) = ˆ x2 i/Ns2 . For two atomic spatial modes, the operahD ˆ x = P (N ˆe,σ − N ˆo,σ ) = P P (−1)i n tor D ˆ i,σ where σ σ i e/o denotes even/odd sites. If ODW 6= 0 the translation symmetry of the atoms in the lattice is spontaneously broken. When light scattering is different from 90◦ other possibilities can arise [18], where additional light-induced spatial modes emerge. To estimate the existence of the pair superfluid state (SFη ), we use the notion of off-diagonal long range order (ODLRO) [33]. This establishes, that given the knowledge of the two-body reduced density matrix ρ2 , if the maximum of its eigenvalues λ scales as the size of the system, then the system has ODLRO. Thus, the presence of SFη is estimated via Oη = 4λ/Ns together with the condition of having

0 6 0 = 0 6= 0

6= 0 0 0 6= 0

0 0 6 0 = 0

off-diagonal elements in ρ2 comparable with the system size. The use of this order parameter has been discussed in the context of the analogue superconducting states in electronic systems [39, 40], and for Fulde-Ferrell-LarkinOvchinnikov (FFLO) states in the context of extended Hubbard models [41]. The correspondence between the quantum phases (QP) of system and the order parameters is in the table (I). In Fig.2 we show how the formation of quantum phases can be controlled by properly orientating the linear polarizer of the probe beam, while modulating the system doping. For each doping realization, the quantum phases can be accessed by fixing the interspecies interaction U , and the effective matter-light coupling strength geff . In Fig.2 (a), for geff /|U | = −0.75, we observe an AFM phase emerging at half-filling by polarizing the probe beam along the y axis (θ = π/2). When doping away from halffilling, ρ = 1 ± N2s , with the external polarizer oriented at θ = 3π/8, a PDW phases arise. PDW phase is characterized by the coexistence of DW and superfluid order. Finally, a homogenous SFη phase occurs when the system is doped further away from half-filling and the orientation of the polarizer is decreased to θ = π/4. The shaded regions in Fig.2 indicate the quantum many-body phase of the atoms, as well as, the values of θ and ρ. As can be seen from Fig.2(b), where geff /|U | = −1, it is possible to produce a different combination of boundaries by changing the angle of the linear polarizer. For instance, one can control the formation of a DW insulator by setting the polarization angle equal to θ = 3π/10 at half-filling, or induce magnetic order by simultaneously increasing the polarization angle and changing the doping of the chain (ρ = 1 ± N2s ). This AFM is achieved by selecting at the same time θ = 2π/5. Further away from half-filling, the attractive on-site interaction (U < 0), inhibits AFM order and gives rise to pair superfluidity, SFη . Indeed, it is remarkable that engineering these quantum phases involves only the simultaneous change of doping and the polarization angle of the probe beam, while maintaining all other parameters fixed. In a typical experiment with ultracold atoms, one could investigate the trajectories and the transitions between states, as well as, their reversibility [4], i.e. at fixed density and by just changing the polarization angle. Understanding these results is possible by analyzing the phase diagram of the system as a function θ for geff < 0, while maintaining fixed U in the strong attractive interaction limit (t/|U |  1). For |geff |/|U |  1, the

4

FIG. 3: (Color online) Phase diagram of fermions in QOL with balanced population. We show the competition of QP for different doping in the strongly attractive interacting limit |U |  t. We plot the QP in the system for several values of doping as a function of the detuning in terms of geff and the angle θ. Doping is: ρ = 1− N4 s

(a), ρ = 1 − N2 (b), and ρ = 1 (c). As the system is doped with s holes [(c) to (a)] the regions in the phase diagram that support a DW insulator and AFM shrink while the intermediate PDW phase enlarge. Depending on the value of geff one can access trajectories such as the ones portrayed P in Fig.2. Parameters are: Ns = 10, t = 0.1, N↑ = N↓ , Nσ = h i n ˆ i,σ i.

ground of the system has broken translational symmetry for θ < π/4, generating a DW insulator, see Fig.3. On the other hand, a paired AFM phase appears for θ > π/4. For intermediate values of |geff |/|U | the competition between short- and long-range interactions becomes evident. The inter-species on-site interaction favors on-site pairing with ODLRO. This competes with the long-range cavity mediated interaction favoring either density wave or magnetic order. For finite |geff |/|U |, the DW↔AFM transition takes place always at angles larger than θ = π/4. When the chain is doped, see Fig.3 (a) and (b), magnetic and density order remain. In the limit where the effective light-matter interaction dominates |geff |/|U |  1, DW or AFM domains surrounded by holes or double occupied sites, depending on θ, are formed. The existence of holes or pairs depends on the way in which the system is doped, sub-doping (supra-doping) the system produces holes (doubly occupied sites) that act as borders between DW or AFM domains. For θ < π/4 and |geff |/|U |  1, the ground state is conformed by a superposition of states with DW domains of the form |0, 0, |DW|1 , 0, 0, .., |DW|2 , 0..., |DW|3 ..0i with holes/pairs acting as walls. Namely, multiple domains with long-range density order result from longrange interaction instead of a single domain |DW| Analogously, for θ > of length ρ = 1 ± Nns . π/4 and |geff |/|U |  1 the ground state is composed of a superposition of states with AFM domains |0, 0, |AFM|1 , 0, 0, ..|AFM|2 , 0..., |AFM|3 ..0i. The length of those domains depends on the value of the doping and tend to maximize the OM . When favouring DW order, away from AFM states there is a smooth transition towards the SFη at fixed |geff |/|U | as θ increases. On the contrary, the transition from a state with DW and/or η-pairing to the AFM state is always sharp, there is no intermediate phase. This occurs as η-pairing and DW are naturally orthogonal to AFM order. An intermediate PDW phase always occurs as the system transits to

the SFη . The coexistence region between DW and SFη phases becomes larger as we move the system away from half-filling, see Fig.3(b) for ρ = 1 − N2s . These magnetic and density modulated spatial structures, could be investigated with methods similar to those employed to study long range hidden magnetic order in doped chains using quantum microscopy. Moreover, in higher dimensions, we can anticipate that the competition between different kinds of domains could play a role in the formation of resonance valence bond (RVB) states [42] and other dimerized phases of quantum matter [43]. Experimental conditions recently achieved in ultracold Fermi atoms are a suitable scenario to reproduce the physics depicted in this Letter. There, 6 Li atoms in one dimensional lattices composed of 7-15 sites are possible [28], with typical tunneling amplitudes of t/h = 400Hz, lattice constant a = 1.15µm and interspecies interaction energy U/h = 2.9 kHz. Since the Feshbach resonance of 6 Li allows the system to have a 3D negative scattering length, an effective attractive interaction in the chain can be achieved [44]. On the other hand, the experimental progress in to have atoms with an external OL in a high-Q cavity allows for typical detunings in the range -70MHz . ∆c /2π . 20MHz with cavity decay rates of κ/2π ∼2MHz. Indeed, it is possible to reach the limit κ  ∆c [4], with 87 Rb atoms (bosonic) in two dimensional OL’s in a single mode cavity and an OL wavelength of 785nm (lattice spacing a = λ/2). The emergence of AFM order and DW order can be measured with single site resolution using quantum-gas microscopes [45, 46], or alternatively, by measuring the ˆ †D ˆ polarization of the output photons. Since nph ∼ hD θ θ i, it is possible to directly access either order parameter OM or ODW by the proper choice of θ. Similarly, DW order has been measured in [4]. Other measurement schemes with light [2, 47] or atomic probing [48] are also possible. We have shown how to control the emergence and engineer the competition of quantum many-body phases with attractive fermions inside an optical cavity illuminated by a transverse pump beam. In particular, we demonstrated that the phases that emerge DW, PDW, AFM and SFη , depend on angle that defines the linear polarization of the pump beam while modifying value of pair/hole doping. This allows full external parametric control, just dependent on atomic loading. In summary, the system studied here allows us to explore the competition between many-body phases of quantum matter with a fully controlled mechanism. This provides a rich experimental landscape for developing new quantum simulators possessing common features with condensed matter systems. This fosters further experimental and theoretical studies in the higher dimensional version of our system. Ultracold fermions in quantum optical lattices could be used with the aim of investigating possible superconducting mechanisms and the interplay with emergent orders of quantum matter. Other possibilities to explore in the future include manipulation of spin order or charge order [49, 50], analog striped

5 superfluidity control [51], the interplay with disorder [25] and non-trivial band topology [52].

CICESE-INAOE-CIO in PIIT, Apodaca N.L. and IFUNAM for their hospitality. ACG acknowledges CONACYT scholarship.

Acknowledgments

SFCB acknowledges financial support from C´atedras CONACYT project 551. SFCB also thanks Consorcio

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Supplemental Material I.

MODEL DETAILS

Our system has a single mode cavity with frequency ωc far from the atomic resonance ωa characterized by a decay rate κ. The atomic spontaneous emission of the atoms Γ is much smaller than the detuning among the pumping mode of light ωp and the atomic frequency ωa . The atoms couple to the cavity via the effective coupling coefficients gp,σ for each spin polarization and the atomic detunning is ∆a = ωp −ωa . The total Hamiltonian of the system in the pump frame of refence is given by, ˆ=H ˆa + H ˆf + H ˆ af H

(6)

ˆ f , the part corresponding with the atomic dynamics H ˆ a , as well as, the interacto the photons in the cavity H tion between the photons in the system and the atoms ˆ af . The atomic dynamics is mediated by the cavity H governed by the standard Hubbard Hamiltonian  XX † X ˆ f = −t H fˆiσ fiσ + H.c. + U n ˆ i↑ n ˆ i,↓ , (7) σ hi,ji

i

† where fˆi,σ , fˆiσ denote the annihilation and creation operators of fermions with spin σ at site i, U is the interspecies on-site interaction strength, and t is the tunnelling amplitude in the single band approximation. The light part ˆ a = −∆c a of the Hamiltonian is H ˆ† a ˆ with ∆c = ωp − ωc being the cavity-pump detunning while we have included the dispersive shift in its definition. The light-matter interaction coupled by the cavity is given by X ∗ ˆ af = H (gp,σ a ˆc Fˆσ† + gp,σ a ˆ†c Fˆσ ), (8) σ

where gp,σ = gc gp ap,σ /∆a is the effective two photon Rabi frequency for each spin polarization, ap,σ are the coherent pump amplitudes for each spin polarization, with gp and gc the light-matter coupling coefficients of the cavity and the pump modes, while the spatial projection of pump and cavity mode onto the atoms is given by, Z ˆ Fσ = d ~r u∗c (~r)up (~r)ˆ nσ (~r), (9)

7 with u∗c,p (~r) the cavity and pump mode functions, and n ˆ σ (~r) the density field of the atoms for each spin projection. It is convenient to expand the atomic fields in terms of the Wannier functions for well localized atoms, such ˆ σ , where D ˆ σ = P Jii n that Fˆσ ≈ D ˆ i,σ is the diagonal i coupling of light to on-site densities. In terms of the Wannier functions of the atoms in the classical optical lattice w(~r − ~rj ) [2], Z Ji,j = d~r w(~r − ~ri )u∗c (~r)up (~r)w(~r − ~rj ). (10) In a one dimensional optical lattice and considering standing waves as mode functions for the light modes, we have uc (~r) = cos(~kc · ~r + φc ) and up (~r) = cos(~kp · ~r + φp ). Thus, the structure of the Jij coefficient can be controlled by properly selecting the orientation of the cavity with respect to the optical lattice and the angle of incidence of the probe beam. We consider ~kp = kp eˆz , and ~kc = π/aˆ ex , (λ = 2a). The pump and the cavity are at 90◦ with respect to each other and the cavity is in the same plane as the classical optical lattice. Therefore, we have X ˆ σ = JD D (−1)i n ˆ i,σ , (11) i

with JD = F[W0 ] a cos(φ0 ) cos(φ1 ), where F[W0 ](~k) is the Fourier transform of W0 (~r) = w2 (~r). The phases φc,p can be chosen arbitrarily, for simplicity we consider φc,p = 0. For a classical optical lattice with depth V0 ∼ 10ER , typically JD ∼ 1. The operators concerning the light modes are given by a ˆc for the cavity mode while ap,R/L is a classical coherent pump describing the light polarization in the circular polarization basis (L or R). The pump mode in the (L, R) basis for an arbitrary orientation θ can be written as
π

1 ap,θ = √ (ap,L eiθ + ap,R e−iθ ), 2

(12)

with ap,R/L 6= 0. Note that ap,θ=0 is the linear polarization in -x while ap,θ=π/2 corresponds to -y polarization. In ˆc = P the steady state of light, the cavity mode is a C D , where C = g /(∆ + iκ). Equivalently, usσ σ σ p,σ c σ ˆ θ, ing L and R polarization basis, we can write a ˆ c = Cθ D with Cθ = g˜p /(∆c + iκ) and g˜p = gc gp ap,θ /∆a . Which is a valid parametrization except from the points where ap,↑ = 0 or ap,↓ = 0, having only one of the spin polarizations coupled. The pump beam intensity |ap,θ |2 is fixed. In the limit where the cavity back-action is dominant and measurement back-action is negligible (∆c  κ) in

the steady state of light [36], the effective matter Hamiltonian can be written as follows, ˆ eff = H ˆ f + geff (D ˆ †D ˆ +D ˆ D ˆ† H θ θ ), 2Ns θ θ where the effective coupling strength is given by

(13)

∆c |˜ gp |2 |JD |2 Ns , ∆2c + κ2

(14)

X (−1)j (ˆ nj,↑ eiθ + n ˆ j,↓ e−iθ ).

(15)

geff = and ˆθ = D

j

Moreover, in terms of linear polarizations we can write, ˆ †D ˆ ˆ ˆ† D 1 + cos(2θ) ˆ 2 1 − cos(2θ) ˆ 2 θ θ + Dθ Dθ = Dx + Dy . (16) 2 2 2 ˆx = D ˆ θ=0 and D ˆ y = −iD ˆ θ=π/2 . with D From the above is clear that one can arbitrarily select depending on the polarization angle how the system can favor or suppress either AFM (θ = π/2), DW (θ = 0) or a combination of both orders (θ 6= 0 or θ 6= π/2) depending on the choice of detuning. Maximal competition between orders is achieved for θ = π/4.

II.

EXACT DIAGONALIZATION CONSIDERATIONS

The Hamiltonian is solved numerically using Exact Diagonalization up to Ns = 10 sites with periodic boundary conditions. The SFη phase concerns the concept of off-diagonal long range order. Due to the finite size of the system we calculate ρ2 (related to the two body reduced density matrix [33]) and compare the maximum of its eigenvalues with the ideal value for a ground state of η-pairs at half-filling. Using this, we define the superfluid order parameter as Oη = 4λ/Ns . The factor of 4 is a normalization, to have for any filling 0 ≤ Oη ≤ 1. Below and above from half-filling, we take into account that deep in the insulating DW phase, ρ2 acquires nonzero off-diagonal elements which are a consequence of the finite size of the system and are irrelevant for true ODLRO. For |geff |/|U |  1 and |geff |/|U |  t/|U | in the limit Ns  1, we have Oη |DW ∼ 4/Ns2 . We compare our numerical results with this estimate to determine emergence of superfluidity. It follows that, pair superfluidity is present in the system if Oη > Oη |DW .