Coherent Two-State Oscillations in False Vacuum Decay Regimes

TL;DR

The paper demonstrates coherent two-state oscillations in the false vacuum decay regime of the one-dimensional transverse-longitudinal-field Ising model, emerging near resonances $h \approx 2J/n$ with a dominant coupling between the false vacuum $|\Omega\rangle$ and symmetric single-bubble states $|S_n\rangle$. A third-order Schrieffer-Wolff transformation in the symmetric subspace yields an effective two-state Hamiltonian exhibiting a collective $\sqrt{L}$ enhancement of the oscillation frequency, i.e., a superradiant-like tunneling splitting $\Delta E \propto \sqrt{L}\,g^3$. The authors identify bubble-size blockade as a mechanism stabilizing coherence for $n\gtrsim L/2$ and show that long-range Ising interactions, as well as cavity-mediated global-range spin squeezing, can lift degeneracies in the multi-bubble manifold and preserve coherent recurrences in large systems. These results provide a practical route to engineer and control recurrences in metastable many-body dynamics, with potential applications in quantum simulations and information processing. The work broadens the landscape of non-perturbative coherent dynamics beyond conventional false vacuum decay and highlights design principles for sustaining coherence in interacting quantum many-body systems.

Abstract

Coherent two-state oscillations are observed in numerical simulations of the one-dimensional transverse-longitudinal-field Ising model (TLFIM) within false vacuum decay regimes. Starting from the false vacuum (a nearly fully polarized ferromagnetic state), we show that in moderate-sized systems, at resonances $h\approx 2J/n$ (with longitudinal field $h$, transverse field $J$, and an integer $n$), the expected decay can give way to coherent oscillations between the false vacuum and a symmetric resonant state. The oscillation frequency, i.e., the tunneling splitting, is observed notably to exhibit a superradiant-like $\sqrt{L}$ enhancement, as confirmed by a Schrieffer-Wolff analysis. In large chains, coherence remains for $n\gtrsim L/2$ due to bubble-size blockade and is robust against stronger transverse fields; for small $n$, long-range interactions can stabilize the oscillations by lifting multi-bubble degeneracies, establishing a robust many-body coherence mechanism beyond perturbative and finite-size limits.

Figures (14)

• Figure 1: (a) The minimal return probability $\min \left(P_{\text{ret}}(t) \right)$ from the false vacuum state $\vert \Omega \rangle$ for $L=8$, shown as a function of the Hamiltonian parameters $h$ (x-axis) and $g$ (y-axis). (b) The same as in (a), but for the sub-leading overlap $P_{\text{sub}}$ between the false vacuum state $\vert \Omega \rangle$ and the eigenstates of the Hamiltonian (\ref{['eq.H']}). The blue regions mark the parameters supporting coherent 2-level dynamics. (c) $P_{\text{sub}}$ at fixed $g=0.8J$ for $L=8,12,16,$ and $20$. (d)--(f) Schematic evolution of the spectrum with increasing $h$, revealing the three dynamical regimes highlighted in (c). (g) Overlaps $P_l$ between the false vacuum state and the low-lying eigenstates $|E_l\rangle$ ($l=0,1,2$) for $L=100$ with $g=0.9J$.
• Figure 2: The evolution starting from the false vacuum state $\vert\Omega\rangle$ at $L=8$, $h=0.67$, and $g=0.11$. (a) The return probability to the initial state. (b) The ZZ correlator. (c) The five translationally invariant states closely support the oscillation. (d) The simulated and the SWT-predicted oscillation period $T$ for various $g$.
• Figure 3: The sub-leading overlap between the false vacuum state and the eigenstates of the Hamiltonian (\ref{['eq.H']}) for $L=12$ around different RBS orders $n$.
• Figure 4: Dynamics with global-range interaction of strength $\beta$. (a) The dynamics of an $L=100$ ring around RBS order $n=3$. The Loschmidt echo of the state at $g=0.1$, $\beta=0$, as $h$ approaches the resonant value for $n=3$. The red curve denotes the resonant $h$, while the blue curves represent the different nearby $h$ approaching this resonance. The system undergoes decay dynamics at resonance. (b) Same as (a), but for $\beta=0.2$. As $h$ approaches resonance, the system stays in two-state oscillation. (c) Scaling of the tunneling splitting $\Delta E=2\pi/T$ with respect to $g$ and $L$ at RBS order $n=3$, extracted from real-time evolution. Data are shown for $g=0.1,0.2$ and $L=16,50,100$ at $\beta=0.2$. Dashed lines are guides to the eye with the expected superradiant slope $1/2$, anchored at the $L=50$ data point.
• Figure S1: State tracking for an $L=16$ ring at $g=0.5$. (a) The maximal overlap $P_{\max}(h)\equiv \max_l|\langle E_l(h)|\Omega\rangle|^2$ stays close to $1$ in each static regime. Across the avoided crossing, the maximizing index $l^\ast(h)\equiv \arg\max_l|\langle E_l(h)|\Omega\rangle|^2$ shifts from $l$ to $l+1$. (b) The lowest 15 energy levels of the post-quench Hamiltonian. After each (avoided) crossing, the eigenstates hosting the dominant overlap with $|\Omega\rangle$ switches. The colormap encodes the overlap of the eigenstates with the false vacuum, $|\langle E_l|\Omega\rangle|^2$.