Quantum Phases of Two-Component Bosons with Spin-Orbit Coupling in Optical Lattices

Abstract

Ultracold bosons in optical lattices are one of the few systems where bosonic matter is known to exhibit strong correlations. Here we push the frontier of our understanding of interacting bosons in optical lattices by adding synthetic spin-orbit coupling, and show that new kinds of density- and chiral-orders develop. The competition between the optical lattice period and the spin-orbit coupling length – which can be made comparable in experiments – along with the spin hybridization induced by a transverse field (i.e., Rabi coupling) and interparticle interactions create a rich variety of quantum phases including uniform, non-uniform and phase-separated superfluids, as well as Mott insulators. The spontaneous symmetry breaking phenomena at the transitions between them are explained by a two-order-parameter Ginzburg-Landau model with multiparticle umklapp processes. Finally, in order to characterize each phase, we calculated their experimentally measurable crystal momentum distributions.

PACS numbers: 67.85.-d,67.85.Hj,67.85.Fg

The physics of spin-orbit coupling (SOC), which links the spin and momentum degrees of freedom in quantum particles, is ubiquitous in nature, ranging from the microscopic world of atoms, such as Hydrogen, to macroscopic solid materials, such as semiconductors. Recently, the effects of SOC have been explored in condensed matter physics in connection with topological insulators [1], as well as with topological superconductors [2], and superconductors without inversion symmetry [3]. In these naturally occurring systems, it is very difficult to control the magnitude of SOC and yet more difficult to study correlated bosons. However it is now possible to create controllable artificial SOC for trapped ultracold fermionic and bosonic atoms [4–9], the physics of which was recently analyzed theoretically in the continuum limit [4, 10–13]. One of the emerging frontiers in this broad area of physics is the interplay of the spin-orbit and lattice characteristic lengths, which can be made comparable in optical lattice systems, where additional contributions from a Zeeman field and strong local interactions also play an important role.

In this Letter, we obtain first the ground-state phase diagrams for two-component (↑, ↓) bosons in the presence of artificial SOC, an effective Zeeman field (created from Rabi coupling and detuning), and local interactions. With zero detuning, we identify four phases: uniform, non-uniform and phase-separated superfluids, along with Mott insulating phases, depending on interactions. Secondly, we develop a Ginzburg-Landau theory for further characterizing these phases. Lastly, we calculate their crystal momentum distributions, which can be compared with experiments.

To describe the quantum phases of two-component bosons with SOC, we begin by introducing the independent particle Hamiltonian

$${\widehat{\mathcal{H}}}_0=\sum _{\mathit{k}}\left(\begin{array}{cc}\hfill {\widehat{b}}_{\mathit{k}\uparrow }^{†}\hfill & \hfill {\widehat{b}}_{\mathit{k}\downarrow }^{†}\hfill \end{array}\right)\left(\begin{array}{cc}\hfill {ϵ}_{\mathit{k}\uparrow }-\mu \hfill & \hfill \hslash \Omega ∕2\hfill \\ \hfill \hslash \Omega ∕2\hfill & \hfill {ϵ}_{\mathit{k}\downarrow }-\mu \hfill \end{array}\right)\left(\begin{array}{c}\hfill {\widehat{b}}_{\mathit{k}\uparrow }\hfill \\ \hfill {\widehat{b}}_{\mathit{k}\downarrow }\hfill \end{array}\right)$$ (1)

in momentum space. Here, ϵ_ks = −2t[cos(k_x + sk_T) + cos k_y + cos k_z] + sħδ/2 for a three-dimensional (3D) optical lattice and k_T = (k_T, 0,0) is the SOC momentum. The length scale 2π/k_T is of the order of the optical lattice spacing a, chosen to be one. The operator ${\widehat{b}}_{\mathit{k}s}^{†}$ describes a creation of s ∈ {↑,↓} ≡ {+, −} boson with momentum k. In addition, the chemical potential μ tunes the average particle density $\rho ={\rho }_{\uparrow }+{\rho }_{\downarrow }\equiv {\sum }_{\mathit{k}s}\langle {\widehat{b}}_{\mathit{k}s}^{†}{\widehat{b}}_{\mathit{k}s}\rangle ∕M$ with M being the number of lattice sites. In cold-atom experiments, the effective Zeeman energy $\Omega \cdot \widehat{\mathbf{F}}$ with Ω = (Ω, 0, δ) and $\widehat{\mathbf{F}}$ being the total angular momentum operator for spin-1/2 has two parts: spin flips through the Rabi frequency Ω and a Zeeman shift via the detuning δ. The Hamiltonian above can be engineered in the laboratory either through Raman processes [4, 5, 14] or via radio-frequency chips [15, 16].

The diagonalization of ${\widehat{\mathcal{H}}}_0$ gives two energy branches

$$E_{\mathit{k}\pm }=\left({ϵ}_{\mathit{k}\uparrow }+{ϵ}_{\mathit{k}\downarrow }-2\mu \pm \sqrt{{({ϵ}_{\mathit{k}\uparrow }-{ϵ}_{\mathit{k}\downarrow })}^2+{(\hslash \Omega )}^2}\right)∕2.$$

For δ = 0 and small ħΩ/t, the lower branch E_k− has two degenerate minima at k_x ≈ ±k_T and k_y = k_z = 0. The two minima approach as the Rabi frequency (spin-hybridization) Ω is increased, and eventualy they collapse into a single minimum at k = 0 when ħΩ/t ≥ 4 sin k_T tan k_T. This double-minimum structure, the introduction of a new length scale 1/k_T and the interactions between particles

$${\widehat{\mathcal{H}}}_{\mathrm{int}}=\frac{1}{2}\sum _{\mathit{kq}}\sum _{s{s}^{\prime }}{U}_{s{s}^{\prime }}{\widehat{b}}_{\mathit{k}s}^{†}{\widehat{b}}_{k+q{s}^{\prime }}^{†}{\widehat{b}}_{k-q{s}^{\prime }}{\widehat{b}}_{\mathit{k}s}$$ (2)

provide additional contributions that are absent in the standard spinless Bose-Hubbard system [17]. In this work, we explore the special case where the same spin repulsions U_ss are nearly identical (U_↑↑≈U_↓↓ = U), but the opposite spin repulsion is different from U, that is, U ≥ U_↑↓ = U_↓↑ ≥ 0. For instance, in the case of a mixture of the m_F = 0 (↓) and m_F = −1 (↑) states from the F = 1 manifold of ⁸⁷Rb, these repulsions are nearly identical (U_↑↑ ≈ U_↓↓ ≈ U_↑↓) [18].

We begin our analysis of the quantum phases of this complex system by investigating first the regime of weak repulsive interactions. In the semi-classical regime (U ≪ tρ), the bosonic fields ${\widehat{b}}_{\mathit{k}s}$ can be written as ${\widehat{b}}_{\mathit{k}s}={\sum }_q^{\prime }\sqrt{M}{\psi }_{\mathit{qs}}{\delta }_{\mathit{k}=(q,0,0)}+{\widehat{a}}_{\mathit{k}s}$, where $\sqrt{M}{\psi }_{\mathit{qs}}$ and ${\widehat{a}}_{\mathit{k}s}$ describe the Bose-Einstein condensate (BEC) with momentum k = (q, 0,0) and the residual bosons outside the condensate, respectively. Considering the single and double minima features of E_k− within the first Brillouin zone, we allow for multiple BECs with different momenta and take the sum ${\sum }_q^{\prime }$ to be over the set of possible momenta {q} along the (k_x, 0, 0) direction. The energy per site of the condensates is

$$\begin{gathered} \frac{E_0}{M}=\sum _q^{\prime }(\begin{array}{cc}\hfill {\psi }_{q\uparrow }^{\ast }\hfill & \hfill {\psi }_{q\downarrow }^{\ast }\hfill \end{array})\left(\begin{array}{cc}\hfill {ϵ}_{\mathit{k}\uparrow }-\mu \hfill & \hfill \hslash \Omega ∕2\hfill \\ \hfill \hslash \Omega ∕2\hfill & \hfill {ϵ}_{\mathit{k}\downarrow }-\mu \hfill \end{array}\right)\left(\begin{array}{c}\hfill {\psi }_{q\uparrow }\hfill \\ \hfill {\psi }_{q\downarrow }\hfill \end{array}\right) \\ +\sum _{\left\{q_i\right\}}^{\prime }\left[\frac{U}{2}\sum _s{\psi }_{q_{1}s}^{\ast }{\psi }_{q_{2}s}^{\ast }{\psi }_{q_{3}s}{\psi }_{q_{4}s}+U_{\uparrow \downarrow }{\psi }_{q_1\uparrow }^{\ast }{\psi }_{q_2\downarrow }^{\ast }{\psi }_{q_3\downarrow }{\psi }_{q_4\uparrow }\right], \end{gathered}$$ (3)

where the sum ${\sum }_{\left\{q_i\right\}}^{\prime }$ is over momenta q_i satisfying momentum conservation q₁ + q₂ = q₃ + q₄ [mod 2π].

After minimization of Eq. (3) with respect to ψ_qs and {q}, we find four different ground states as shown in Fig. 1(a) for the weak-coupling regime with parameters U = t/ρ, U_↓↑ = 0.9U, and k_T = 0.2π. In the superfluid phases (SF_±), the set of BEC momenta {q} consists of a single value ($\stackrel{‒}{q}$ > 0 in SF₊ and $-\stackrel{‒}{q}$ < 0 in SF₋) since the detuning δ tilts the single-particle spectrum and lifts the degeneracy of the double minima in E_k−. In these “single-q” states, the particle density is uniform, while the phase of the condensate spatially varies with pitch vector (±$\stackrel{‒}{q}$, 0, 0). In the striped superfluid (ST) phase for relatively small ħΩ/t, a BEC is formed with two different momenta $-{\stackrel{‒}{q}}_1$ and ${\stackrel{‒}{q}}_2$ due to a double-minimum dispersion in E_k−. The interference of these two momenta leads to a non-uniform density profile along the x direction, resulting in a stripe pattern. Moreover, the scattering process under momentum conservation q₁ + q₂ = q₃ + q₄ with $q_3=q_4=-{\stackrel{‒}{q}}_1$ and $q_2={\stackrel{‒}{q}}_2$ (or vice-versa) gives rise to a higher harmonic component with $q_1=-2{\stackrel{‒}{q}}_1-{\stackrel{‒}{q}}_2$ (or $q_1={\stackrel{‒}{q}}_1+2{\stackrel{‒}{q}}_2$). Similar processes generate higher harmonics with interval ${\stackrel{‒}{q}}_1+{\stackrel{‒}{q}}_2$, thus making the set {q} have a large number of different momenta $-{\stackrel{‒}{q}}_1+n({\stackrel{‒}{q}}_1+{\stackrel{‒}{q}}_2)$, where n is an integer.

FIG. 1:

(color online). Ground-state properties in the weak-coupling regime with U/t = 1/ρ and U_↓↑ = 0.9U. (a) Phase diagram of detuning δ versus Rabi frequency Ω for k_T = 0.2π. The thick red (thin black) curves denote first- (second-) order transitions and the black dots indicate multicritical points. In the δ > 0 (δ < 0) region to the left side of the dash-dotted line, the SF₋ (SF₊) exists only as a metastable state. (b) Roton-like softening in the elementary excitations for quasi-momentum k = (k_x, 0, 0) and ħΩ/t = 0.4. We set ħδ/t = 0.4 (in SF₊) for the dotted lines and ħδ/t = 0.06 (at the SF₊-ST boundary) for the solid lines. (c) The k_T dependence of the ground state when ρ_↓ = ρ_↑ (δ = 0). The yellow and darker green regions limited by the black-dashed and red lines are the CSF and period-locked ST phases illustrated in (e). (d) The plateaux in $\stackrel{‒}{q}$ of the CSF and period-locked ST phases as a function of k_T for ħΩ/t = 1.0. The dashed lines denote the width of dominant plateaux with commensurate wavenumber $\stackrel{‒}{q}$ (e) Density (the size of dots) and chiral (the direction of arrows) patterns in the commensurate phases.

When ħΩ/t is large, the SF₊ and SF₋ phases are continuously connected at δ = 0 through the conventional superfluid (SF₀) with zero-momentum BEC. However, when ħΩ/t has intermediate values, a direct first-order transition from SF₊ to SF₋ takes place, and thus the spin population difference ρ_↓ − ρ_↑ exhibits a sudden jump from positive to negative. Therefore, in the experimental situation where the population of each spin is balanced (ρ_↓ = ρ_↑), the system is unstable against spatial phase separation (PS) of spin-down-rich SF₊ and spin-up-rich SF₋ states.

The quadratic part of the Hamiltonian in terms of ${\widehat{a}}_{\mathit{k}s}$, ${\widehat{\mathcal{H}}}_B={\sum }_{\mathit{k}}{\widehat{\mathit{a}}}_{\mathit{k}}^{†}{H}_{\mathit{k}}^{(2)}{\widehat{\mathit{a}}}_{\mathit{k}}$, is a generalized Bogoliubov Hamiltonian and includes quantum fluctuations outside the condensate perturbatively. We diagonalize ${\widehat{\mathcal{H}}}_B$ numerically via a generalized Bogoliubov transformation [1, 20], and obtain the spectrum of elementary excitations. In Fig. 1(b) we show typical excitation spectra of the SF+ states. We can see a roton-like minimum at a finite quasimomentum with the excitation energy approaching zero as the detuning δ is decreased (increased) as we move from the SF₊ (SF₋) phase towards the ST phase. The transition from SF₊ (or SF₋) to ST is induced by the softening of the roton-like minimum, similar to the standard superfluid-supersolid transition [21]. The momentum of the roton-like excitations largely determines the characteristic reciprocal vector ${\stackrel{‒}{q}}_1+{\stackrel{‒}{q}}_2$ of the ST state resulting from the phase transition. Furthermore, the transition from SF₊ or SF₋ to the ST phase can also be first order as indicated by the red solid line shown in Fig. 1(a). In this case, the energy gap of roton-like excitations jumps discontinuously to zero at the SF_±/ST boundary.

The weak coupling phase diagram shown in Fig. 1(a) reveals ground states which are very similar to those in the continuum limit [4, 10–13], where the band structure due to the optical lattice is not important. However, the phase diagram of SOC momentum k_T/π versus ρΩ/t at ρ_↓=ħ_↑, shown in Fig. 1(c), illustrates the remarkable competition between the intrinsic reciprocal vector of the underlying optical lattice and characteristic vector ${\stackrel{‒}{q}}_1+{\stackrel{‒}{q}}_2$ of the ST phase. In the spin symmetric case (ρ_↓ = ρ_↑), the two wavevectors ${\stackrel{‒}{q}}_1$ and ${\stackrel{‒}{q}}_2$ are equal, that is, ${\stackrel{‒}{q}}_1={\stackrel{‒}{q}}_2\equiv \stackrel{‒}{q}$ leading to ${\stackrel{‒}{q}}_1+{\stackrel{‒}{q}}_2=2\stackrel{‒}{q}$. The phase diagram of k_T/π versus ħΩ/t in the range of k_T = π to 2π is exactly the same as that of Fig. 1(c) since the lattice Hamiltonian ${\widehat{\mathcal{H}}}_0+{\widehat{\mathcal{H}}}_{\mathrm{int}}$ is invariant under the gauge tranformation ${\widehat{b}}_{\mathit{k}s}\to {\widehat{b}}_{\mathit{k}+(\pi ,0,0)s}$, as easily verified by direct substitution.

In Fig. 1(d), when k_T is nearly commensurate to the lattice reciprocal wavenumber 2π, such as k_T ≈ π/4, 2π/3, and π/2, the pitch vector $\stackrel{‒}{q}$ of the ST state spontaneously takes an exact commensurate value over a finite range of k_T. As a result, the curve of $\stackrel{‒}{q}∕\pi$ versus k_T/π exhibits multiple plateaux in the ST phase. This effect can be attributed to umklapp processes q₁ + q₂ − q₃ − q₄ = 2πn with nonzero integer n that contribute to lower the energy of the system. In particular, when k_T ≈ π/2, BEC occurs with only two momenta $\pm \stackrel{‒}{q}=\pm \pi ∕2$ since all the higher-harmonics momenta are reduced to ±π/2 due to the Brillouin zone periodicity. In this special case where $\stackrel{‒}{q}∕\pi =1∕2$, the interference of the two momenta does not lead to striped density pattern, but to Z₂ chiral symmetry breaking. This state is analogous to the chiral superfluid (CSF) state, which has been discussed in Bose-Hubbard ladders [22–25]. In the present case, the 3D lattices for the two spin components and the Rabi couplings play the role of rails and rungs, respectively, of a synthetic “two-leg ladder” in four (three spatial plus one extra spin) dimensions as illustrated in Fig. 1(e). For other commensurate ST phases, where $\stackrel{‒}{q}∕\pi$ takes an irreducible fraction ζ/η with ζ and η being integers, the superfluid phases break Z_η symmetry, but preserve a stripe pattern in the atom density. The stabilization of these commensurate phases is a specific feature of spin-orbit coupled systems in optical lattices with interactions and are completely absent in interacting continuum systems. Had we illustrated all the possible commensurate/incommensurate transitions in Fig. 1(d), the graph of $\stackrel{‒}{q}∕\pi$ versus k_T/π would have had an infinite number of steps at rational values of $\stackrel{‒}{q}∕\pi$, producing a mathematical function known as the Devil’s staircase.

In addition to the interplay between different length/momentum scales discussed above in the weak coupling regime, another particular feature of lattice systems is the existence of Mott insulator (MI) phases induced by strong interactions and commensurate particle fillings. To describe the Mott physics in the presence of SOC and Zeeman fields, we employ the Gutzwiller variational method [20]. Under the assumption that the ground state is given by a direct product state in real space, the Hamiltonian can be mapped into an effective single-site problem with variational mean fields ${\psi }_{\mathit{is}}\equiv \langle {\widehat{b}}_{\mathit{is}}\rangle$, where ${\widehat{b}}_{\mathit{is}}$ is the annihilation operator of spin-s boson at lattice site i. To deal with the ST phases, we solve simultaneously all inequivalent single-site problems connected via mean fields due to the nonuniformity.

Here, we consider up to 2 × 10³ mean fields ψ_is along the x direction for each spin and thus the momentum resolution is δk_x ~ 0.001π [20], while the y and z directions are assumed to be uniform. In the ST phase, the inhomogeneous state is a result of the length scale introduced by the SOC, while in the absence of SOC a new length scale leading to a supersolid state appears due to long-range interactions [26].

Figure 2 shows phase diagrams in the μ/U-t/U plane for several values of the Rabi frequency Ω in the spin symmetric case ρ↓ = ρ↑(δ = 0). In Fig. 2(a), where there is no hybridization of the two spin components (ħΩ/t = 0), the phase boundaries of the MI lobes are identical to those in the absence of SOC [27] since the gauge transformation ${\widehat{b}}_{\mathit{k}s}\to {\widehat{b}}_{\mathit{k}+s{\mathit{k}}_{T}s}$ eliminates the momentum transfer k_T from the problem. The even-filling Mott transitions become first order in a two-component Bose-Hubbard model for large inter-component repulsions (for example, U_↓↑ ≳ 0.68U when ρ = 2) [27–30]. In the superfluid phase outside the Mott lobes for k_T ≠ 0, the spin-down and spin-up bosons independently form the SF₊ state with $\stackrel{‒}{q}=k_T$ and SF₋ with $-\stackrel{‒}{q}=-k_T$, respectively.

FIG. 2:

(color online). Ground-state phase diagrams in the (t/U,μ/U) plane, obtained by the Gutzwiller self-consistent calculations for different values of ħΩ/t. We set the other parameters as U_↓↑ = 0.9U, k_T = 0.2π, and ρ_↓ = ρ_↑ (δ = 0).

The phase diagrams displayed in Figs. 2(b-d), illustrate the effects of increasing ħΩ. When the Rabi frequency Ω is non-vanishing, the two spin components mix, forming a nonuniform ST state with two opposite momenta $-\stackrel{‒}{q}$ and $\stackrel{‒}{q}$ and their associated higher harmonics, analogous to the stripe phase in continuum systems. Figure 2(b) shows that the transition from the odd-filling MI to the ST phase occurs via an intermediate SF₀ state. A direct transition to the ST state occurs only for very small ħΩ/t (not shown: ħΩ/t ≲ 0.04 for ρ = 1). As seen in Figs. 2(c-d), when the value of ħΩ/t is increased, the SF₀ phase also emerges near the tip of the ρ = 2 MI lobe, and eventually joins other SF₀ regions. The SF₊ and SF₋ states only phase separate for small fillings ρ ≲ 1 and a very narrow region around the ρ = 2 MI lobe for large ħΩ/t.

To see the interplay between local correlations and spin mixing, we plot in Figs. 3(a-c) phase diagrams of U/t versus ħΩ/t for fixed density ρ = 2. As shown in Fig. 3(a), large spin hybridization Ω mixes the two spin components, and destabilizes the ST state. As seen in Figs. 3(b-c) the transition between the MI and ST state is discontinuous (first-order) for any ħΩ/t when opposite spin repulsion U_↓↑/U is large, but for small U_↓↑/U, the transition is continuous. In order to clarify this effect, we develop next a Ginzburg-Landau theory.

FIG. 3:

(color online). Nonuniform superfluid-insulator transitions at ρ = 2 for (a) U_↓↑ = 0.9U and (b) U_↓↑ = 0.2U. We set k_T = 0.2Π and ρ_↓ = ρ_↑ (δ = 0). The vertical dashed line in (a) marks ħΩ/t = 0.72. The enlarged view of the region indicated by the dashed box in (b) is shown in (c). The fourth-order Ginzburg-Landau coefficients and the value of $\stackrel{‒}{q}$ along the MI transition line of (c) are plotted in (d).

The nature of the superfluid-insulator transition when ρ_↑ = ρ_↓ can be described by the Ginzburg-Landau energy

$$\frac{E_{\mathrm{GL}}}{M}=\xi (\mathit{k})\left({\Phi }_{\mathrm{I}}^2+{\Phi }_{\mathrm{II}}^2\right)+\frac{{\Gamma }_1}{2}\left({\Phi }_{\mathrm{I}}^4+{\Phi }_{\mathrm{II}}^4\right)+{\Gamma }_2{\Phi }_{\mathrm{I}}^2{\Phi }_{\mathrm{II}}^2$$ (4)

up to fourth order of the order parameters ${\Phi }_{\mathrm{I}}=\mid {\Phi }_{\stackrel{‒}{q}}\mid$ and ${\Phi }_{\mathrm{II}}=\mid {\Phi }_{-\stackrel{‒}{q}}\mid$, which describe the BEC with $\mathit{k}=(\pm \stackrel{‒}{q},0,0)$.

Note that the higher harmonics are negligible in the vicinity of the transition. The value of $\stackrel{‒}{q}$ is determined so that the function ξ(k) attains its minimum value $-\stackrel{‒}{\mu }$ at $\mathit{k}=(\pm \stackrel{‒}{q},0,0)$. When $\stackrel{‒}{\mu }>0$ > 0, the bosons condense at $\stackrel{‒}{q}$ and/or $-\stackrel{‒}{q}$ with $\stackrel{‒}{q}$ ≠ 0. For Γ₁ < Γ₂, the minimization of Eq. (4) gives $\mid {\Phi }_{\stackrel{‒}{q}}\mid \ne 0$ and $\mid {\Phi }_{-\stackrel{‒}{q}}\mid =0$ (or vice-versa), and thus the Z₂ symmetry related to $\stackrel{‒}{q}$ or $-\stackrel{‒}{q}$ is broken. In this case, the transition from MI to PS takes place. On the other hand, the condition Γ₁ >Γ₂ gives $\mid {\Phi }_{\stackrel{‒}{q}}\mid =\mid {\Phi }_{-\stackrel{‒}{q}}\mid \equiv \Phi \ne 0$, resulting in the transition to the ST or CSF phase. When $\stackrel{‒}{q}∕\pi$ is an irreducible fraction ζ/η, the relative phase $\varphi =\mathrm{Arg}({\Phi }_{\stackrel{‒}{q}}∕{\Phi }_{-\stackrel{‒}{q}})$ is determined by the minimization of additional η-particle umklapp process, ${\Gamma }_{\eta }^{\prime }(({\Phi }_{\stackrel{‒}{q}}^{\ast }{)}^{\eta }({\Phi }_{-\stackrel{‒}{q}}{)}^{\eta }+({\Phi }_{-\stackrel{‒}{q}}^{\ast }{)}^{\eta }({\Phi }_{\stackrel{‒}{q}}{)}^{\eta })\propto \cos \eta \varphi$, which still has η-fold degeneracy. Thus the ST transition is associated with U(1) × Z_η symmetry breaking about the global and relative phases of ${\Phi }_{\pm \stackrel{‒}{q}}$.

The coefficients ξ(q), Γ₁, Γ₂ and ${\Gamma }_{\eta }^{\prime }$ are related to the microscopic system parameters in the original Hamiltonian by performing a perturbative expansion based on a direct-product MI state. For the specific relations see supplemental material [20]. We show in Fig. 3(d) the values of Γ₁ and Γ₂ along the line that separates the MI phase from the others as seen in Fig. 3(c). Note that if Γ₁ < 0 for Γ₁ < Γ₂ or Γ₁ + Γ₂ < 0 for Γ₁ > Γ₂, the condensates have a negative compressibility, and the transition becomes first-order.

To assist in the experimental identification of these quantum phases, Fig. 4 shows the crystal momentum distribution $\langle {\widehat{b}}_{\mathit{k}s}^{†}{\widehat{b}}_{\mathit{k}s}\rangle$ at fixed density ρ_↑ = ρ_↓ = 1. For the PS and ST states, we evaluate $\langle {\widehat{b}}_{\mathit{k}s}^{†}{\widehat{b}}_{\mathit{k}s}\rangle$ via the Bogoliubov Hamiltonian ${\widehat{\mathcal{H}}}_B$ at relatively weak interactions. However, for the SF₀ and MI states, we calculate $\langle {\widehat{b}}_{\mathit{k}s}^{†}{\widehat{b}}_{\mathit{k}s}\rangle$ via a generalized Holstein-Primakoff approach [20] based on the Gutzwiller variational state describing the strongly coupled regime. Since the PS state consists of independent domains of SF₊ and SF₋, we plot the simple average of the two contributions. The crystal momentum distribution discussed here does not include the effects of Wannier functions, but can be easily extracted from standard momentum distribution measurements.

FIG. 4:

(color online). The crystal momentum distributions $\langle {\widehat{b}}_{\mathit{k}s}^{†}{\widehat{b}}_{\mathit{k}s}\rangle$ (k = (k_x, 0, 0)) of the four different states along the line of ħΩ/t = 0.72 in Fig. 3(a) (at U/t = 0.5, 1.5, 30, and 44). The contribution from SF₊ (SF₋) in the PS phase is plotted by the solid (dashed) lines.

As seen in Figs. 4, the momentum distribution of the PS state exhibits two independent peaks around k = ($\stackrel{‒}{q}$, 0, 0) and k = ($-\stackrel{‒}{q}$ 0,0), which come from the SF₊ and SF₋ contributions, respectively, while the ST state shows additional peaks due to the higher harmonics. The SF₀ state exhibits a peak around k = 0 as in the case of a standard uniform superfluid state, although the reflectional symmetry with respect to k_x → −k_x ondensation occurs in the neighboring superfluid state. The stark differences between these crystal momentum distributions also enable the direct imaging of the different phases present in inhomogeneous trapped systems.

In summary, we investigated the quantum phases of two-component bosons in optical lattices as a function of spin-orbit coupling, Rabi frequencies and interactions. In phase diagrams at zero detuning, we identified four different regions occupied by uniform, non-uniform and phase-separated superfluids or Mott insulators. Finally, we characterized these phases by calculating their crystal momentum distributions, which can be easily measured experimentally.