Simulating Dynamical Phases of Chiral $p+ip$ Superconductors with a Trapped ion Magnet

Two-dimensional $p+ip$ superconductors and superfluids are systems that feature chiral behavior emerging from the Cooper pairing of electrons or neutral fermionic atoms with nonzero angular momentum. Their realization has been a longstanding goal because they offer great potential utility for quantum computation and memory. However, they have so far eluded experimental observation both in solid-state systems as well as in ultracold quantum gases. Here, we propose to leverage the tremendous control offered by rotating two-dimensional trapped-ion crystals in a Penning trap to simulate the dynamical phases of two-dimensional $p+ip$ superfluids. This is accomplished by mapping the presence or absence of a Cooper pair into an effective spin-1/2 system encoded in the ions’ electronic levels. We show how to infer the topological properties of the dynamical phases, and discuss the role of beyond mean-field corrections. More broadly, our work opens the door to use trapped-ion systems to explore exotic models of topological superconductivity and also paves the way to generate and manipulate skyrmionic spin textures in these platforms.

Popular Summary

Two-dimensional (2D) $p$-wave superconductors are systems featuring exotic behaviors including a chiral superconducting order parameter and nontrivial topology. These attributes make them a promising resource for quantum computation and memory. In particular, the nonequilibrium dynamics of topological superconductors under quenches offers great insight into understanding the superconducting mechanism and the system’s topology. Nevertheless, to date, a clean experimental observation of $p$-wave superconductivity has remained elusive in both solid-state systems as well as ultracold fermionic gases. Here, we leverage the tremendous control offered by trapped-ion systems to design and propose an experiment to explore the nonequilibrium dynamical phases of 2D $p$-wave superconductors. We take advantage of an effective mapping of the fermionic model onto a model of interacting pseudospin-1/2 particles. The two spin states emulate the presence or absence of Cooper pairs of electrons and can be encoded in two electronic levels of the ions. Moreover, the center-of-mass motion of the ions emulates a bosonic molecular channel. In the regime where the latter can be adiabatically eliminated, its role is to effectively mediate virtual $p$-wave interactions between the pseudospins.

Ion crystals stored in Penning traps are rotating when viewed in the lab frame, and here we exploit this unique feature in the context of a quantum simulation application. We explain how the experimental toolbox available to control 2D planar ion crystals in Penning traps can be utilized to initialize spin textures with nontrivial topology, engineer effective $p$-wave interactions, and measure the superconducting order parameter. We show that signatures of all three dynamical phases predicted by mean-field theory can be observed in this simulator. Because of the finite number of ions, our simulator naturally opens the pathway to explore the effects of quantum fluctuations on the mean-field dynamical phases. We further demonstrate how our proposed simulator can be used to infer a winding number that can distinguish dynamical phases with trivial and nontrivial topologies.

Our work demonstrates the potential of trapped-ion crystals to explore the rich topological behaviors in $p$-wave superconductors that have been predicted theoretically but have not yet been observed experimentally. In parallel it also opens a path to the creation and stabilization of skyrmionic spin textures in trapped-ion systems, which can find further applications in studying magnetism and in quantum information processing.

Figure 1

Probing dynamical phases of $p+ip$ superfluids using ion crystals in Penning traps. (a) The fermionic model is mapped on to spins encoded in the internal states of the ions, where spin up (down) represents the presence (absence) of a Cooper pair (Anderson pseudospin mapping). Here we show representative spin textures that can be engineered for the topologically nontrivial BCS and trivial BEC phases. (b) The dynamical phases are classified according to the long-time behavior of the magnitude of an order parameter—phase I, $|\mathrm{\Psi }(t)|\to 0$; phase II, $|\mathrm{\Psi }(t)|\to$ nonzero constant; phase III, $|\mathrm{\Psi }(t)|$ displays persistent oscillations. (c) Schematic of our experimental proposal. State initialization, $p$-wave interactions and readout are all achieved using appropriate parameters for a pair of optical dipole force (ODF) lasers and a pair of copropagating Raman lasers. In contrast to prior implementations, the ODF difference wave vector $\delta \mathbf{k}$ has both an out-of-plane and in-plane component (see side view). The result is a tilted traveling-wave lattice that crosses the crystal plane slightly obliquely and thus couples the ions’ electronic degrees of freedom, the out-of-plane center-of-mass mode, and the in-plane crystal rotation. The spin-space directions $\mathcal{Z}$ in panel (a) and $Z$ in panel (c) are related by a rotation, as discussed in Sec. 3.

Figure 2

Manifestation of dynamical phases in a 200 ion crystal. Initial BCS-like spin textures (top panels) and time evolution of $|\mathrm{\Psi }(t)|$ in phases (a) I, (b) II, and (c) III. In (a) $K/J=10$, while in (b) and (c) $K/J=1$. The dynamics of $|\mathrm{\Psi }(t)|$ are computed both using mean-field theory and the discrete truncated Wigner approximation (DTWA) method. The DTWA results show that the finite size of the crystal leads to a decay of $|\mathrm{\Psi }(t)|$ even in phases II and III. However, the buildup of quantum correlations in these phases is captured by a second order parameter $\tilde{\mathrm{\Psi }}$ [Eq. (20)]. In all cases, the decay of $|\mathrm{\Psi }(t)|$ in the absence of interactions is plotted for reference. Crystal parameters are detailed in Appendix pp2 and chiral spin states are initialized according to Appendix pp7.

Figure 3

Inferring topological properties of the dynamical phases. (a) Initial BEC-like spin texture. (b) Mean-field, long-time winding number of the effective magnetic field texture as the relative strength of the single-particle ($K$) and interaction ($J$) terms are varied. In the strong interaction case ($K/J<0.6$), the texture is BEC-like and topologically trivial, while in the weak interaction case ($K/J>0.6$), it is BCS-like and topologically nontrivial. Representative field textures are shown, where each arrow now indicates the unit vector of the effective magnetic field acting at that site, and the color code indicates the normalized $B_{\mathcal{Z}}$ component. (c),(d) The sense of rotation of the order parameter in the complex plane reveals the winding number of the underlying magnetic field texture. The topologically trivial (nontrivial) dynamical phase is associated with a counterclockwise (clockwise) rotation of $\mathrm{\Psi }(t)$. While mean-field predicts a stable limit cycle (gray dashed line), DTWA calculations (colored circles) show the order parameter spiraling in towards the origin at long times (plotted here until $Jt=100$). The color gradient indicates the arrow of time. (e) The Cooper pair distribution function (CPDF) displays an even (odd) number of zero crossings in the topologically trivial (nontrivial) case. These features are preserved even at times $Jt\sim 100$, when the order parameter has decayed considerably due to quantum fluctuations. For comparison, the dashed lines show the CPDF computed using mean-field theory. Crystal parameters are detailed in Appendix pp2 and chiral spin states are initialized according to Appendix pp7.

Figure 4

Realizing the two-channel model. The drumhead c.m. mode plays the role of the molecular channel in the two-channel $p$-wave model. As the coupling to the c.m. mode is tuned from off resonant (${\delta }_1^2/G^2\gg 1$) to near resonance (${\delta }_1^2/G^2\sim 0$), its effects can be clearly observed in the time evolution of $|\mathrm{\Psi }(t)|$. Here, we fix $B_1=G/\sqrt{10}$ and vary ${\delta }_1$ to obtain the different curves. Crystal parameters are detailed in Appendix pp2 and chiral spin states are initialized according to Appendix pp7.

Figure 5

Impact of off-resonant interactions on the one-channel model dynamics for crystals formed under two different trapping parameters: (a) case A and (b) case B (see Appendix 13). In both cases, we progressively add terms to the pure one-channel model and study their impact. We add the following terms in the specified order: Chiral spin exchange by all modes ($J_{12}$), antichiral spin exchange by all modes ($J_{11}$), and achiral spin exchange by all modes ($J_5$). Here, we start from a BCS-like initial state [top panel of Fig. 2] and use the experimental parameters discussed in Appendix pp2. We have used $B_1\approx J$, where the value of $J/(2\pi )\approx 405\mathrm{Hz}$.

Figure 6

Benchmarking the DTWA method. Solid lines are computed by numerical propagation of the Schrödinger equation, while the dotted lines are obtained using the DTWA method. Here, we assume that the crystal is made of perfectly concentric rings of ions and that the spin state is initialized in a BCS-like state [top panel of Fig. 2]. Here, we have used the ratio $K/J=1$.

Figure 7

Demonstration of Delaunay triangulation for a crystal with 200 ions. The triangulation is used to identify triplets of neighbors on the discrete lattice for computing the winding number of the spin texture ($Q$) or that of the effective magnetic field texture ($W$).