Quantum Simulation of the Cooperative Jahn-Teller Transition in 1D Ion Crystals

The Jahn-Teller effect explains distortions and nondegenerate energy levels in molecular and solid-state physics via a coupling of effective spins to collective bosons. Here we propose and theoretically analyze the quantum simulation of a many-body Jahn-Teller model with linear ion crystals subjected to magnetic field gradients. We show that the system undergoes a quantum magnetic structural phase transition which leads to a reordering of particle positions and the formation of a spin-phonon quasicondensate in mesoscopic ion chains.

Figure 1

Quantum magnetic structural phase transition with trapped ions. (a) Normal state $(g<g_{\mathrm{c}})$. (b) Quantum magnetic zigzag phase ($g>g_{\mathrm{c}}$). For moderate chain lengths long-range zigzag order appears together with an antiferromagnetic spin phase.

Figure 2

Mean-field results with $N=20\mathrm{ions}$. Energy units such that ${\omega }_z=1$, $\Delta =2.2$. (a) Phonons per site in the chiral base. Squares: homogeneous chain with $g=0.25$, $0.3$, $0.35$ and $t_{j,j+1}=0.5$. Circles: Coulomb chain with $g=0.4$, $0.45$, $0.5$, and distances scaled such that at the center, $t_{10,11}^{\mathrm{Coul}}=0.5$. In (b) and (c) we show results for a Coulomb (continuous line) and an homogeneous (dashed line). (b) Vibrational energies of the radial collective modes. (c) Mean number of phonons as a function of $g$. The phase transition point at $g\sim 0.25$ and 0.3 is visible for the Coulomb and homogeneous chain, respectively.

Figure 3

Quantum Gaussian fluctuations $F_{\delta a_{\gamma ,n}}$. Continuous, dash-dotted, and dashed lines correspond to $\gamma =\mathrm{s}$ (spin waves), $l$ ($l$ phonons) and $r$ ($r$ phonons), respectively. We have considered homogeneous chains and units such that ${\omega }_z=1$, $\Delta =2.2$, $t_{j,j+1}=0.5$ in homogeneous chains. (a) $N=20\mathrm{ions}$ and different $g$ values. As expected fluctuations exhibit a peak at the critical point. Note that $F_{\{a_{\mathrm{s},n}\}}$ and $F_{\{a_{\mathrm{r},n}\}}$ are enhanced since spin and $r$-phonon operators are closer to resonance in (4). (b) $g=0.3$ and different values of $N$. Fluctuations diverge with growing $N$ in accordance with the Mermin-Wagner theorem [21].

Figure 4

Comparison for $\mathrm{O}.\mathrm{P}.=\sum _{j,k}\sum _{ϵ=r,l}⟨a_{ϵ,j}^{†}{a}_{ϵ,k}⟩/N^2$ defined in the text in a short-range homogeneous CJT chain with $t=0.2$, and units such that $\Delta =2$, ${\omega }_z=1$, and $N=20$.