Scalable phonon-laser arrays with self-organized synchronization

Abstract

Quantum mechanical oscillators operating at frequencies up to the GHz regime have been predicted to support phonon lasing -- self-sustained coherent vibrational motion emerging when the effective gain exceeds intrinsic losses. Current phonon-laser proposals face two key limitations, namely: they lack scalability and rely on coupling all oscillators to a common field, which significantly restricts flexibility and prevents selective, on-demand phonon lasing at specific locations. Given that numerous applications and theoretical insights naturally emerge from scalable many-body systems, addressing these limitations is timely. In this Letter, we demonstrate how scalable arrays of individually addressable phonon lasers can be generated through local driving in a quantum many-body Ising-like spin chain. We rigorously establish the resonance conditions under which mechanical oscillators transition from thermal motion to sustained coherent self-oscillation. Unlike previous approaches that rely on a common coupling bus, our proposal employs purely local driving, resulting in an inherently modular and scalable architecture ideally suited for integration into large-scale quantum systems. Additionally, our approach enables on-demand lasing of individual mechanical oscillators at specific sites by simply switching the spin-mechanical coupling interaction on and off, provided specific resonance conditions are satisfied. Notably, our phonon laser array is robust against resonance mismatches and naturally exhibits both pairwise self-organized synchronization and global phase locking near resonance. Finally, we outline an experimental implementation within current experimental capabilities.

Figures (6)

• Figure 1: Mechanical oscillators (MO) individually coupled to sites of a spin chain of length $N$. Local time-dependent nearest-neighbour exchange couplings at precise resonance conditions enable steady-state phonon-laser arrays. Our lasing scheme allows us to easily decouple MOs at any site (shown as semi-transparent oscillators in the sketch), thereby enabling the generation of arbitrary on-demand arrays of steady-state phonon lasers.
• Figure 2: Case of two spins, with only one spin coupled to a MO at resonance condition $\Omega_1=4\overline\omega+\omega_1$\ref{['eq_resonance_conditions']}. (a) Comparison of the time evolution of phonon amplification, characterized by the average phonon number $\langle\hat{b}_1^\dagger \hat{b}_1\rangle$, for the full Hamiltonian (solid green line) and the effective Hamiltonian (dashed black line). Inset: shows the steady-state average phonon number $\langle\hat{b}_1^\dagger \hat{b}_1\rangle_\infty$ as a function of $\Omega_1/\overline\omega$. (b) Second-order correlation function $g^{(2)}(0)$ versus time $\overline\omega t$. Inset: steady-state $g^{(2)}(0)$ as a function of $\Omega_1/\overline\omega$, and Wigner distribution $W(x,p)$. We compare full Hamiltonian dynamics (solid green) with effective Hamiltonian results (dashed black). (c) Min-max normalized steady-state phonon population versus $J_1/\overline\omega$. (d) Power spectrum $S(\omega)$ as a function of $J_1/\overline\omega$. Frequency defined with respect to the MO frequency $\omega_1$. Other parameters are: $\Delta_1=\Delta_2=2\overline\omega$, $\omega_1=5\overline\omega$, $\Omega=9\overline\omega$, $\lambda_1=0.4\overline\omega$, $J_1=0.08\overline\omega$, $\Gamma_{1}=\Gamma_{2}=2\times10^{-2}\overline\omega$, $\gamma_1=8\times10^{-4}\overline\omega$, $\bar{n}_{1}^{s}=\bar{n}_{2}^{s}=0.01$ and $\bar{n}_{1}^{m}=0.1$.
• Figure 3: Spin-mechanical array of $N=10$ sites, with MOs placed only at selected sites $j=(1,3,4,6,8,9)$. (a) Phonon expectation value $\langle \hat{b}_j^\dagger \hat{b}_j \rangle$ as a function of time $\overline\omega t$. (b) Second-order correlation function $g^{(2)}(0)$ of each MO as a function of time $\overline\omega t$ under the same resonance conditions as in (a). (c) Kuramoto parameter and (inset) phase difference for some pairs of lasing MOs. Other parameters are: $\omega_j/\overline\omega=\{8.0, 8.0, 7.9995, 7.9994, 8.0, 7.9992, 8.0, 7.999, 7.9989, 7.9988\}$, $\lambda_j/\overline\omega = \{0.4,0,0.42,0.38,0,0.41,0,0.37,0.43,0\}$, $J_j/\overline\omega=0.2$, $\Omega_j/\overline\omega=4$, $\Gamma/\overline\omega=8\times10^{-2}$, $\gamma/\overline\omega=10^{-3}$, $\bar{n}_{j}^{s}=0.01$ and $\bar{n}_{j}^{m}=0.1$.
• Figure S1: (a) Time evolution of phonon amplification quantified by the average phonon number $\langle \hat{b}_1^\dagger \hat{b}_1 \rangle$ at the resonance condition $\Omega_1 = 4\overline\omega$. The inset shows the corresponding steady-state value $\langle \hat{b}_1^\dagger \hat{b}_1 \rangle_{\infty}$ as a function of $\Omega_1/\overline\omega$. (b) Second-order correlation function $g^{(2)}(0)$ versus dimensionless time $\overline\omega t$ at the same resonance. Inset: Wigner distribution $W(x,p)$ evaluated at $\Omega_1/\overline\omega = 8$. Solid green lines denote the full Hamiltonian dynamics, while dashed black lines correspond to the effective Hamiltonian, $\hat{\mathcal{H}}_{II}^{\textit{eff}}$. Other parameters are: $\Delta_1=\Delta_2=2\overline\omega$, $\omega_1=8\overline\omega$, $\lambda_1=0.4\overline\omega$, $J_1=0.1\overline\omega$, $\Gamma_{1}=\Gamma_{2}=8\times10^{-3}\overline\omega$, $\gamma_1=10^{-3}\overline\omega$, $\bar{n}_{1}^{s}=\bar{n}_{2}^{s}=0.01$ and $\bar{n}_{1}^{m}=0.1$.
• Figure S2: (a) Phonon expectation and (b) second order correlation function vs time for $N=10$ mechanical oscillators. Here, we observe that lasing generation is stimulated for those mechanical oscillators where the resonance condition $\Omega_j = \sum_{j}^{j+1}\Delta_{k} + \omega_k$ is fulfilled. Other mechanical oscillators remain decoupled from the spin chains and can be activated on demand. The parameters (in units of $\overline\omega$) are: $\Delta_k=2$, $\omega_j=\{5, 0, 7, 0, 9, 0, 12, 0, 16, 0\}$, $\Omega_j=\{4+\omega_1, 4+\omega_3, 4+\omega_3, 4+\omega_5, 4+\omega_5, 4+\omega_7, 4+\omega_7, 4+\omega_9, 4+\omega_9, 0\}$, $\lambda_j = \{0.4,0,0.4,0,0.4,0,0.4,0,0.4,0\}$, $J_j=0.3$, $\Gamma_{j}=8\times10^{-2}$, $\gamma_j=10^{-3}$.