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

TL;DR

The paper proposes a cavity QED quantum simulator that maps 2D topological BCS superconductors onto a lattice of atomic pseudospins, using incommensurate cavity-lattice wavelengths to engineer momentum-dependent p- and d-wave pairing channels. By tuning drive parameters and photon-mediated interactions, the authors realize competing $p_x+ip_y$ and $d_{x^2-y^2}+id_{xy}$ orders, and provide a mean-field protocol for ground-state preparation with continuous, non-destructive readout via cavity photons. They analyze both equilibrium and non-equilibrium dynamics, revealing regimes of single-order dominance, coexistence, and rich dynamical phases, including topological transitions detected through order parameter phases and dynamical Chern numbers. The framework enables direct exploration of topological transitions and order competition in regimes challenging for solid-state and ultracold-atom systems, with implications for understanding Majorana modes and non-equilibrium topology in engineered quantum matter.

Abstract

Realizing and controlling the unconventional pairing featured by topological superconductors remains a central challenge. We introduce a cavity QED quantum simulator that engineers competing chiral $p_x+ip_y$ and $d_{x^2-y^2}+id_{xy}$ orders by tailoring cavity-mediated couplings between atomic pseudospins that emulate momentum-dependent pairing channels. The desired spatially inhomogeneous cavity-mediated couplings can be engineered in a 2D optical lattice using incommensurate cavity-lattice wavelengths naturally occurring in cavity QED systems. This minimal and fully tunable platform enables controlled state preparation and continuous measurement of superconducting order parameters, revealing phases in both equilibrium and sudden-quench settings with a single dominant pairing channel, as well as coexistence regimes with competing pairing channels. Crucially, our implementation allows direct observation of topological transitions in and out of equilibrium, providing a powerful route to the quantum simulation of competing topological superconducting phases that remain elusive in solid-state and ultracold-atom systems.

Figures (10)

• Figure 1: Mapping topological BCS superconductors to cavity QED simulators. (a) We consider Anderson pseudospin mapping between the presence/absence of a Cooper pair and pseudospin states $|\uparrow\rangle_{\mathbf{k}}$/$|\downarrow\rangle_{\mathbf{k}}$, and then relate the Anderson pseudospins in 2D momentum space (top panel) to the atomic spins pinned in 2D lattice sites inside a cavity QED simulator (bottom panel). We can thus implement the momentum dependence of pairing interactions via the spatial inhomogeneity of spin exchange interactions, which is generated by the standing-wave structure of the cavity mode (blue) and the running-wave structure of the laser drive (red). The $x,y$ directions of the lattice are mapped to magnitude (indicated by green color) and phase (indicated by magenta color) of momentum respectively. (b) The pseudospin textures for different topological phases (characterized by Chern number $Q$) in topological BCS superconductors: $d$-BCS phase ($Q=2$), $p$-BCS phase ($Q=1$), and BEC phase ($Q=0$). See Sec. \ref{['sec:topology']}-B for the definition of Chern number $Q$.
• Figure 2: Cavity QED setup for the BCS model with both $p_x+ip_y$ and $d_{x^2-y^2}+id_{xy}$ pairings. (a) We consider atomic spins pinned in 2D lattice sites inside a standing-wave optical cavity, and apply external laser drives (red color, drive A from the side and drive B along the cavity) to couple $|\uparrow\rangle$ to $|e\rangle$ (see the left inset). The standing wave cavity mode $b$ (blue color) couples $|\downarrow\rangle$ to $|e\rangle$ and mediates spin exchange interactions between atoms (see the right inset). Another cavity mode $r$ (light red color) only supports external drive B and does not lead to interatomic interactions. The dynamics of superconducting order parameters $\Delta_p(t)$ and $\Delta_d(t)$ can be observed by continuously tracking the light leaking out of the cavity. For state preparation, we engineer a Raman transition via an additional drive $C$ coupling $|\downarrow\rangle$ to $|e\rangle$ (see App. \ref{['sec:cavity']} for details). (b) Frequency diagram for engineering interaction Hamiltonian $\hat{H}_{\rm cav}$, including atomic transitions ($\omega_e$), external laser drive $A$ and $B$ ($\omega_{p,A}$, $\omega_{p,B}$), cavity modes ($\omega_c$) and emitted photons (see text). For clarity, external drive $C$ only used in state preparation is not shown in this diagram. (c) Proposed experimental sequence for probing sudden quench dynamics of topological superconductors. We prepare the mean-field ground state aligned to self-consistent field $\mathbf{B}^{\rm self}_{\mathbf{n}}(\chi_{p,i},\chi_{d,i})$ (see Eq. (\ref{['eq:meanh']})) by ramping external field $\mathbf{B}^{\rm ext}_{\mathbf{n}}(t')$ (see Eq. (\ref{['eq:drive']})). We then let the system evolve under self-consistent field $\mathbf{B}^{\rm self}_{\mathbf{n}}(\chi_{p,f},\chi_{d,f})$.
• Figure 3: Dynamical phases and topological phase transitions for $p$-wave pairing only. (a) Dynamical phase diagram of suddenly quenching the interaction strength from $\chi_{p,i}$ to $\chi_{p,f}$. We fix the number of Cooper pairs to $N_C/N=0.35$. The solid lines mark the dynamical phase boundary, and the dashed line marks the condition $\chi_{p,i}=\chi_{p,f}$ for equilibrium physics. The circles and triangles mark the position of the curves in (b) and (c) on the phase diagram. (b) Examples of the three dynamical phases (I, II ,and III). The yellow lines describe phase I dynamics at $\chi_{p,i}N/J=4$, $\chi_{p,f}N/J=1$, blue lines are for phase II at $\chi_{p,i}N/J=2$, $\chi_{p,f}N/J=4$, and purple lines are for phase III at $\chi_{p,i}N/J=1$, $\chi_{p,f}N/J=3$. In both (b) and (c), the curves with lighter color include Hamiltonian dynamics only (see Eq. (\ref{['eq:caveff']})), while the ones with darker color include dissipation due to cavity photon loss with $\kappa/|\delta_{c,A}|\,=2\times 10^{-3}$. (c) Examples of the II-BCS phase and the II-BEC phase. The blue lines are for $\chi_{p,f}N/J=4$ and the orange lines are for $\chi_{p,f}N/J=8$. The left panel shows the equilibrium case with $\chi_{p,i}=\chi_{p,f}$, and the right panel shows quench dynamics from $\chi_{p,i}N/J=2$. In both panels, we show trajectories of $\Delta_p$ with $Jt/2\pi\in[0,2]$. (d) The long-time chemical potential $\mu_{\infty}$ as a function of $\chi_{p,f}$. We only include Hamiltonian dynamics for the evaluation of $\mu_{\infty}$.
• Figure 4: Competing $p$-wave and $d$-wave orders in equilibrium. (a) Equilibrium phase diagram for Eq. (\ref{['eq:caveff']}) with a fixed number of Cooper pairs $N_C/N=0.35$. The dashed lines separate three regimes based on the stability of mean-field eigenstates: $p+ip$ regime (only $p_x+ip_y$ pairing is stable, yellow color), $d+id$ regime (only $d_{x^2-y^2}+id_{xy}$ pairing is stable, green color), coexistence regime (both states are stable, shaded area). The black solid line marks the first order phase transition between the $p_x+ip_y$ and the $d_{x^2-y^2}+id_{xy}$ pairing. The pink line marks the topological transition between $p$-BCS and BEC phases of the $p+ip$ regime. The $d+id$ regime within the range of this diagram is in the $d$-BCS phase. (b,c) Stability of mean-field eigenstates. The system is prepared in a $p_x+ip_y$ eigenstate at (b) $\chi_p N/J=3$, $\chi_d N/J=8$ and (c) $\chi_p N/J=3$, $\chi_d N/J=4$. The lines with lighter color include Hamiltonian dynamics only (see Eq. (\ref{['eq:caveff']})). The lines with darker color include dissipation due to cavity photon loss ($\kappa/|\delta_{c,A}|\,=2\times 10^{-3}$, $\delta_{c,A}\approx \delta_{c,B}$). The system is stable against small initial $d$-wave pairing ($\varepsilon_d=10^{-2}$) in (c), but unstable in (b). (d) Schematics for the stability of $p_x+ip_y$ pairing with fixed $\chi_p$. As we increase $\chi_d$, in the $p+ip$ regime the system has a single minimum for $p$-wave pairing, in the coexistence regime the system has two local minima for $p$-wave and $d$-wave pairing respectively. While in the $d+id$ regime the system has a single minimum for $d$-wave pairing, so $p_x+ip_y$ pairing becomes unstable.
• Figure 5: Competing $p$-wave and $d$-wave orders away from equilibrium. (a) We prepare the $p_x+ip_y$ eigenstate at $\chi_{p,i} N/J=1$ and $N_c/N=0.35$, and perform a sudden quench of the interaction strength to $\chi_{p,f}N/J$. The dynamical phases of the quench dynamics is characterized by the long-time standard deviation of $|\Delta_p|$ (see text). For the case of $\chi_d=0$ (red), we can identify three dynamical phases (phase III, phase III* and phase II-BEC) shown in Fig. \ref{['fig:dpt']}(a). For the case of $\chi_d N/J=3.5$ (blue), a small initial $d$-wave pairing ($\varepsilon_d=10^{-2}$) can lead to significant changes in the quench dynamics. (b) Oscillation frequency $f_{\rm osc}$ of phase III/III* dynamics as a function of $\chi_{p,f}$. The transition between phase III and phase III* is indicated by a frequency dip in the case of $\chi_d=0$. Additional frequency kinks are found in the case of $\chi_dN/J=3.5$. (c) Examples of quench dynamics ($\chi_{p,f}N/J=4.0,5.6,6.0$ from left to right). The purple (green) lines describe dynamics of $|\Delta_p|/J$ ($|\Delta_d|/J$) at $\chi_{d,f}N/J=3.5$, and the lines with lighter colors describe dynamics at $\chi_{d,f}N/J=0$. The instability of $|\Delta_d|$ gives rise to an additional quench of the system and can be visualized in $|\Delta_p|$ dynamics. All the numerical results in this figure do not include dissipative effects.