Simulation of topological superconductors and their competing orders using photon-mediated interactions

The gist The cavity QED quantum simulator engineers competing chiral px + ipy and dx2−y2 + idxy orders by tailoring cavity-mediated couplings between atomic pseudospins, providing a minimal, fully tunable platform to study competing topological superconducting phases.

Topological Superconductors and Competing Orders

Topological superconductors intertwine nontrivial topology with many-body physics, hosting protected Majorana zero modes with prospects for fault-tolerant quantum computation <ref:2512.17889#pg2>. Realizing such phases remains difficult because they require unconventional Cooper pairing with nonzero orbital angular momentum (e.g. px±ipy, dx2−y2 + idxy), which is typically much weaker than conventional s-wave pairing <ref:2512.17889#pg2>. The paper proposes a cavity QED quantum simulator that realizes and probes competing chiral px + ipy and dx2−y2 + idxy superconducting orders <ref:2512.17889#pg2>. This platform provides a minimal, fully tunable, and pristine setting in which multiple unconventional orders can be engineered simultaneously, enabling direct access to the long-standing competition between pairing channels that remains unresolved in solid-state and ultracold-atom systems <ref:2512.17889#pg2>.

Cavity QED Implementation

The system uses an ensemble of N atoms trapped in a 2D square optical lattice coupled to a standing-wave optical cavity featuring two degenerate cavity modes <ref:2512.17889#pg4>. The central idea is to shape the spatial profile of the atom-light coupling in real space so that the resulting photon-mediated spin exchanges inherit the orbital structure of unconventional px + ipy and dx2−y2 + idxy pairing symmetries <ref:2512.17889#pg5>. This emulation is accomplished using incommensurate wavelengths between cavity modes and the optical lattice that naturally arise in cavity QED systems <ref:2512.17889#pg2>. The desired spin exchange interactions are mediated by four-photon exchange processes where both the excited state e⟩ and the cavity mode b only virtually participate in the dynamics <ref:2512.17889#pg5>. The effective Hamiltonian for this system, after adiabatic elimination of the excited state e⟩, is given by Eq. (5) <ref:2512.17889#pg5>.

Preparing Mean-Field Ground States

The mean-field ground state of the BCS model can be prepared by engineering a Raman transition via an additional drive C coupled to the cavity mode b <ref:2512.17889#pg6>. This preparation involves ramping external drive parameters for a time tprep to prepare the ground state of Eq. (A36) which is equivalent to the MF ground state of the interaction Hamiltonian <ref:2512.17889#pg6>. The goal is to adiabatically ramp the drive parameters for a time tprep, which allows access to both equilibrium phases and out-of-equilibrium dynamics <ref:2512.17889#pg6>.

Probing Dynamical Phases and Topology

The non-equilibrium quench dynamics of the BCS model can be characterized by three types of dynamical phases (I, II and III) <ref:2512.17889#pg5>. These dynamical phases are described using a time-averaged or steady-state order parameter that shows non-analytic change at the dynamical phase boundaries <ref:2512.17889#pg5>. The paper demonstrates how to resolve all dynamical phases and the associated topological transitions for pure px + ipy interactions <ref:2512.17889#pg6>. Furthermore, it reveals regimes in which a single order dominates as well as coexistence regimes where both px + ipy and dx2−y2 + idxy orders are stable <ref:2512.17889#pg6>.

Probing Competing Orders

The platform can access the long-standing problem of competing unconventional superconducting orders by introducing the d-wave interaction channel and examining its interplay with the p-wave channel <ref:2512.17889#pg5>. The stability of states against small perturbations is determined by the interplay between χp and χd <ref:2512.17889#pg5>. In equilibrium, a coexistence regime exists where both pairing channels can be stable, leading to a first-order phase transition between px+ipy and dx2−y2 + idxy pairings <ref:2512.17889#pg5>. Away from equilibrium, competing orders modify the dynamical phases; for instance, a finite d-wave interaction can play a non-negligible role for phase III/III∗ quench dynamics <ref:2512.17889#pg5>.

Conclusion and Outlook

This approach establishes a versatile route to study order competition, dynamical phase structure, and topological phase transitions in regimes inaccessible to solid-state materials or conventional ultracold gases <ref:2512.17889#pg6>. The platform can be extended to resolve full spin textures, carry out radio-frequency–style spectroscopy using an auxiliary atomic level <ref:2512.17889#pg6>, and to explore pairing competition involving multiple pairing channels <ref:2512.17889#pg6>. By trapping two fermionic atoms per lattice site and mapping ↑⟩ → cˆ†n,σcˆ†n,00⟩, the simulator can directly incorporate fermionic degrees of freedom, providing a route to investigate emergent boundary effects and to search for Majorana zero modes in a fully controlled setting <ref:2512.17889#pg6>.

How it works

1. The momentum dependence of pairing interactions is implemented via the spatial inhomogeneity of spin exchange interactions generated by the standing-wave structure of the cavity mode (blue) and the running-wave structure of the laser drive (red) <ref:2512.17889#pg3>.

2. Momentum space pairing physics is encoded in real-space internal atomic degrees of freedom, enabling controlled engineering of various pairing channels and avoiding cooling motional levels below quantum degeneracy <ref:2512.17889#pg5>.

3. The continuous and non-destructive monitoring of order parameters is achieved by tracking photons leaking out from cavity mode b, as the coherent state amplitude αb(t) is directly determined by the BCS order parameters <ref:2512.17889#pg8>.

4. Topological phase transitions in and out of equilibrium can be detected by tracking the phase of BCS order parameters, which allows for the determination of non-equilibrium topological Chern numbers W <ref:2512.17889#pg5>.

Topological Phase Transitions

The equilibrium topological phase transition between p-wave and d-wave pairing is characterized by the Chern number Q, where Q = 2π/4π Z ∫dkdϕ ∂⟨⃗σk⟩/∂k × ∂⟨⃗σk⟩/∂ϕ · ⟨⃗σk⟩ <ref:2512.17889#pg2>. The topological phase transition within phase II is marked by an isolated root u = 0 in the equation L 2(u) = 0 <ref:2512.17889#pg3>. The separation between phase III and phase III∗ is set by a doubly-degenerate, negative real root pair for L 2(u) = 0 <ref:2512.17889#pg4>.

Dynamical Phase Boundaries

The dynamical phase boundaries are determined by the change in the number of isolated pairs of roots in the equation L 2(u) = 0 <ref:2512.17889#pg10>. The transition between phase I and phase II is marked by a solution u0 that depends on sgn(β), indicating a strong-to-weak quench when sgn(β) > 0 and a weak-to-strong quench when sgn(β). The transition between phase II and phase III/III∗ is also determined by the sign of sgn(β) <ref:2512.17889#pg10>.

Stability Analysis

The stability boundary for the px + ipy solution is given by χd,stab = Xnη4n2q(Jη2n−µ)2 + ∆2p,0η6n/4q(Jη2n−µ)2 + ∆2p,0η3/2 <ref:2512.17889#pg2>. The stability boundary for the dx2−y2 + idxy solution is given by χp,stab = Xnη2n2q(Jη2n−µ)2 + ∆d,0η6n/4q(Jη2n−µ)2 + ∆d,0η4n3/2 <ref:2512.17889#pg2>. The stability analysis shows that the px + ipy solution is stable if ∆d,0/(∆p,0εd) 1 (large χd) <ref:2512.17889#pg2>. The stability boundary for the dx2−y2 + idxy solution is thus given by Eq. (B44) <ref:2512.17889#pg2>.

Summary of Approximations

Key approximations include the adiabatic elimination of atomic excited states δe ≫ ωp,A − ωp,B, ωp,A − ωc, gc√N, omegaA, omegaB, γ <ref:2512.17889#pg2>. The adiabatic elimination of cavity modes requires that the detunings are much larger than the coupling strengths and decay rates <ref:2512.17889#pg2>. The interference effect between drive A and B is ignored under the condition ωp,A − ωp,B ≫ J, χpN, χdN <ref:2512.17889#pg2>.

Continuous Readout of Order Parameters

The mean-field equation for the coherent state amplitude of cavity mode b is dαb = -κ/2 αb - ie−iδc,AtomegaAgc2/2δeχp∆p − ie−iδc,BtomegaBgc2/2δeχd∆d <ref:2512.17889#pg8>. The number of photons leaking out of the cavity during a time step is given by n¯ = ¯np + ¯nd <ref:2512.17889#pg8>.

1. Improved AI can simulate competing topological superconducting phases by engineering spatially inhomogeneous cavity-mediated couplings between atomic pseudospins to emulate momentum-dependent pairing channels like px + ipy and dx2−y2 + idxy orders.

2. Improved AI can perform continuous, non-destructive monitoring of superconducting order parameters by tracking photons leaking out from cavity mode b to access both equilibrium phases and out-of-equilibrium dynamics.

3. Improved AI can identify topological transitions in and out of equilibrium by observing how small perturbations around equilibrium can induce a drastic destabilization of one order parameter accompanied by the rapid growth of the competing order, and quenches can produce strongly enhanced dynamics.

4. Improved AI can determine non-equilibrium topological phase transitions by tracking the phase of BCS order parameters in the long-time steady state, replacing µ → µ∞ and ∆α,0 → ∆α,∞ in the definition of Q.

5. Improved AI can map momentum space pairing physics onto real-space internal atomic degrees of freedom to engineer various pairing channels and avoid the need for cooling motional levels below quantum degeneracy.

6. Improved AI can perform stability analysis of competing orders by calculating the second-order derivative, d2dt2∆d squared = -2(χd∆p,0εd)2 / (1 - ∆d,0/∆p,0ε).

7. Improved AI can predict the stability boundaries for unconventional pairing channels based on initial conditions and system parameters using analytical results such as χd,stab ≈ 3χp or χd,stab ≈ Jµχp.