Detecting d-wave superfluid and d-density wave states of ultracold Fermions on optical lattices

We propose a pump probe experiment for detecting the d-wave superfluid and d-density wave phases of ultracold Fermions on an optical lattice. The pump consists of periodic modulations of the optical lattice intensity which creates quasiparticle pairs in these systems. The changes in the momentum distribution under the drive can be used to measure quasiparticle dispersion and gap anisotropy. Further, we show that the pattern of peaks and dips in the spin selective density-density correlation function provides a phase sensitive probe of the symmetry of the order parameter in these systems.

💡 Research Summary

In this paper the authors propose a novel pump‑probe scheme for ultracold fermionic atoms in an optical lattice that can unambiguously identify d‑wave superfluid (d‑SF) and d‑density‑wave (DDW) phases—two candidate orders relevant to the physics of high‑Tc cuprates. The “pump” consists of a weak, periodic modulation of the lattice depth, V(t)=V0+δV sin(ωt), which in turn modulates both the nearest‑neighbour tunnelling amplitude J and, through the microscopic pairing mechanism, the pairing gap Δ0. In the linear‑response regime (modulation amplitude λ≪1) this drive creates quasiparticle pairs with total momentum zero and total energy equal to the drive frequency ω.

The authors first treat the d‑SF using a simple two‑dimensional BCS Hamiltonian with a d‑wave order parameter Δk=Δ0(cos kx−cos ky). The resulting quasiparticle dispersion Ek=√(ξk²+Δk²) produces banana‑shaped constant‑energy contours that intersect the Brillouin‑zone diagonals at the nodes (±π/2,±π/2). By solving the time‑dependent Bogoliubov‑de Gennes equations (or, equivalently, performing second‑order perturbation theory in λ) they show that when ω≈2Ek the drive resonantly populates quasiparticles along the corresponding banana. Consequently the momentum distribution n(k)=⟨c†kck⟩ acquires pronounced enhancements precisely on the resonant contour. Scanning ω therefore maps out Ek(k) and directly reveals the anisotropic gap and the presence of nodes: a d‑wave superfluid responds at arbitrarily low ω, whereas an s‑wave superfluid shows a threshold at ω=2Δ0.

Beyond the single‑particle distribution, the authors focus on the spin‑selective density‑density correlator N↑↓(q)=⟨ρ↑(q)ρ↓(−q)⟩, which can be extracted from noise correlations in time‑of‑flight images or measured via elastic light scattering. In the BCS state N↑↓(q) reduces to a sum over k of u_k(t)v*k(t) u*{k+q}(t)v_{k+q}(t). Because the drive concentrates quasiparticles near the banana tips, the correlator exhibits a characteristic pattern of peaks and dips at wave vectors q that connect pairs of tips. If Δk and Δ_{k+q} have the same sign the interference is constructive (peak); if they have opposite sign the interference is destructive (dip). Thus the relative arrangement of peaks and dips provides a phase‑sensitive test of the order‑parameter symmetry, directly analogous to the π‑shift observed in corner‑junction SQUID experiments on cuprates, but now realized in a cold‑atom platform.

The second part of the paper applies the same methodology to the DDW state, where the order parameter appears as a particle‑hole condensate ⟨c†k c_{k+Q}⟩ with Q=(π,π). The mean‑field Hamiltonian doubles the unit cell, yielding two bands that touch at the Brillouin‑zone corners. At half‑filling the lower band is full and the upper empty; the lattice modulation promotes electrons across the gap, again creating particle‑hole pairs with total energy ω. The resulting momentum‑distribution changes again trace banana‑shaped contours, but now the centers of the bananas remain fixed at (±π/2,±π/2) regardless of filling because Q is fixed. By comparing the noise‑correlation patterns for Δk=Δ0(cos kx−cos ky) and for its absolute‑value counterpart Δk=Δ0|cos kx−cos ky|, the authors demonstrate that the peak‑dip arrangement distinguishes genuine d‑wave sign changes from a merely anisotropic but sign‑preserving gap.

Experimentally, the required measurements are within reach: the momentum distribution can be obtained by band‑mapping or standard time‑of‑flight imaging, while N↑↓(q) follows from shot‑noise analysis of real‑space density images or from Bragg scattering. Importantly, the method relies only on the binary presence of a peak versus a dip, making it robust against quantitative renormalizations due to strong correlations.

In summary, the paper introduces a versatile, non‑destructive probe for fermionic quantum simulators. Periodic lattice‑depth modulation provides direct access to quasiparticle dispersions, gap anisotropy, and, crucially, the sign structure of the order parameter. This technique opens a clear pathway to experimentally discriminate d‑wave superfluidity from competing orders such as DDW, and can be extended to other unconventional pairing symmetries in strongly correlated cold‑atom systems.