Dissipation-Induced $d$-Wave Pairing of Fermionic Atoms in an Optical Lattice

We show how dissipative dynamics can give rise to pairing for two-component fermions on a lattice. In particular, we construct a parent Liouvillian operator so that a BCS-type state of a given symmetry, e.g., a $d$-wave state, is reached for arbitrary initial states in the absence of conservative forces. The system-bath couplings describe single-particle, number-conserving and quasilocal processes. The pairing mechanism crucially relies on Fermi statistics. We show how such Liouvillians can be realized via reservoir engineering with cold atoms representing a driven dissipative dynamics.

Figure 1

(a) Symmetry in the $d$-wave state, represented by a single off site fermion pair exhibiting the characteristic sign change under spatial rotations. In a $d$-wave BCS state, this pair is delocalized over the whole lattice. (b),(c) The dissipative pairing mechanism builds on (b) Pauli blocking and (c) delocalization via phase locking. (b) Illustration of the action of Lindblad operators using Pauli blocking for a Néel state (see text). (c) The $d$-wave state may be seen as a delocalization of these pairs away from half-filling (shown is a cut along one lattice axis).

Figure 2

Numerical illustration of the uniqueness of the steady state, showing evolution under the master equation with Lindblad operators from Eq. (1). (a) Entropy computed exactly for four atoms on a $4\times 1$ lattice, showing exponential convergence from a completely mixed state to a pure state. (b) Fidelity to the $d$-wave BCS state, $⟨{\mathrm{BCS}}_N|\rho |{\mathrm{BCS}}_N⟩$ with 4 atoms on a $4\times 4$ grid, computed via a quantum trajectories method (see text). Dashed lines show sampling error.

Figure 3

(a) Level scheme for alkaline earth atoms with $I=1/2$, showing excitation to a metastable level while flipping the nuclear spin, and induced decay by coupling to the ${}^{1}P_1$ level. (b) Illustration of $J_i^{+}$ implementation in 1D: (i) A longer period lattice for ${}^{3}P_0$ identifies a triple of wells, and atoms from the central level are transferred to the ${}^{3}P_0$ manifold with spin flip. (ii) The ${}^{3}P_0$ potential wells are adiabatically split into two; (iii) Decay is induced, returning the atom to the ${}^{1}S_0$ level via coupling to a lossy cavity mode.